task_name,problem_type,instruction,instance,solution,obj,instance_variant,solution_variant,context_index,input_format,input_index_base
SPP,SPP,"I’ve been handed the job of choosing which collection routes the neighborhood’s bin trucks will run so every street gets picked up by one, and only one, route. The idea is to pick a group of those candidate routes that together cover every street exactly once, and to keep the overall operating bill as small as possible — that bill is just the sum of the costs of the routes that get used. The exact streets, routes, and their costs are shown below.

# num_streets=18
# num_candidate_routes=70
route_id,route_operating_cost,streets_covered
S1,48,7 8 9 11
S2,104,1 2 3 4
S3,392,15 16 17 18
S4,65,11 12 13 14 17
S5,207,12 13 14
S6,162,10 12
S7,152,9 10 12 13
S8,230,5 7 8 10 11
S9,246,6 9 10
S10,164,6 8
S11,20,6 8 9 10
S12,420,11 12 13 15 16
S13,384,13 14 15 17
S14,388,11 12 14 18
S15,176,2 3
S16,99,2 3 4
S17,220,1 2 3 4 6
S18,186,9 12
S19,410,6 8 9 10 12
S20,272,5 7 8 9
S21,155,9 10 11 13 15
S22,18,1 3
S23,390,14 15 16 17 18
S24,276,7 8 10
S25,87,9 12 13
S26,58,1 2
S27,86,1 5
S28,140,7 8
S29,106,15 16
S30,106,3 5
S31,156,1 3 6
S32,144,2 3 4 5
S33,21,3 4 5
S34,16,16 18
S35,315,13 14 15 16 17
S36,190,7 9
S37,84,8 11 12 14
S38,154,14 15
S39,146,8 11
S40,344,7 8 9 10
S41,292,12 14 15 16
S42,160,1 2 3 4 5
S43,252,4 6 7
S44,60,14 16 17 18
S45,180,13 15 16 17 18
S46,68,11 12
S47,40,4 6
S48,24,10 11 12 13
S49,400,6 7 8 9 10
S50,16,8 9 10 12
S51,244,5 7 9 10
S52,87,14 16 17
S53,10000,1
S54,10000,2
S55,10000,3
S56,10000,4
S57,10000,5
S58,10000,6
S59,10000,7
S60,10000,8
S61,10000,9
S62,10000,10
S63,10000,11
S64,10000,12
S65,10000,13
S66,10000,14
S67,10000,15
S68,10000,16
S69,10000,17
S70,10000,18

If you want the answer in a machine-friendly way, just stick to this tiny JSON shape when you reply:

{
  ""solution"": [""route_id"", ...]
}

This just means ""solution"" should be an array of the route IDs you pick (one entry per chosen route). Think of it like writing down which collection routes get used — nothing fancy, just the list. This is just a sketch of the shape I need, not the actual selection.

Please make sure you use the exact route identifiers from the instance input — don't rename them or invent new ones. 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 18, 'num_sets': 70, 'density': 0.15555555555555556, 'sets': [{'id': 1, 'elements': [7, 8, 9, 11], 'cost': 48}, {'id': 2, 'elements': [1, 2, 3, 4], 'cost': 104}, {'id': 3, 'elements': [15, 16, 17, 18], 'cost': 392}, {'id': 4, 'elements': [11, 12, 13, 14, 17], 'cost': 65}, {'id': 5, 'elements': [12, 13, 14], 'cost': 207}, {'id': 6, 'elements': [10, 12], 'cost': 162}, {'id': 7, 'elements': [9, 10, 12, 13], 'cost': 152}, {'id': 8, 'elements': [5, 7, 8, 10, 11], 'cost': 230}, {'id': 9, 'elements': [6, 9, 10], 'cost': 246}, {'id': 10, 'elements': [6, 8], 'cost': 164}, {'id': 11, 'elements': [6, 8, 9, 10], 'cost': 20}, {'id': 12, 'elements': [11, 12, 13, 15, 16], 'cost': 420}, {'id': 13, 'elements': [13, 14, 15, 17], 'cost': 384}, {'id': 14, 'elements': [11, 12, 14, 18], 'cost': 388}, {'id': 15, 'elements': [2, 3], 'cost': 176}, {'id': 16, 'elements': [2, 3, 4], 'cost': 99}, {'id': 17, 'elements': [1, 2, 3, 4, 6], 'cost': 220}, {'id': 18, 'elements': [9, 12], 'cost': 186}, {'id': 19, 'elements': [6, 8, 9, 10, 12], 'cost': 410}, {'id': 20, 'elements': [5, 7, 8, 9], 'cost': 272}, {'id': 21, 'elements': [9, 10, 11, 13, 15], 'cost': 155}, {'id': 22, 'elements': [1, 3], 'cost': 18}, {'id': 23, 'elements': [14, 15, 16, 17, 18], 'cost': 390}, {'id': 24, 'elements': [7, 8, 10], 'cost': 276}, {'id': 25, 'elements': [9, 12, 13], 'cost': 87}, {'id': 26, 'elements': [1, 2], 'cost': 58}, {'id': 27, 'elements': [1, 5], 'cost': 86}, {'id': 28, 'elements': [7, 8], 'cost': 140}, {'id': 29, 'elements': [15, 16], 'cost': 106}, {'id': 30, 'elements': [3, 5], 'cost': 106}, {'id': 31, 'elements': [1, 3, 6], 'cost': 156}, {'id': 32, 'elements': [2, 3, 4, 5], 'cost': 144}, {'id': 33, 'elements': [3, 4, 5], 'cost': 21}, {'id': 34, 'elements': [16, 18], 'cost': 16}, {'id': 35, 'elements': [13, 14, 15, 16, 17], 'cost': 315}, {'id': 36, 'elements': [7, 9], 'cost': 190}, {'id': 37, 'elements': [8, 11, 12, 14], 'cost': 84}, {'id': 38, 'elements': [14, 15], 'cost': 154}, {'id': 39, 'elements': [8, 11], 'cost': 146}, {'id': 40, 'elements': [7, 8, 9, 10], 'cost': 344}, {'id': 41, 'elements': [12, 14, 15, 16], 'cost': 292}, {'id': 42, 'elements': [1, 2, 3, 4, 5], 'cost': 160}, {'id': 43, 'elements': [4, 6, 7], 'cost': 252}, {'id': 44, 'elements': [14, 16, 17, 18], 'cost': 60}, {'id': 45, 'elements': [13, 15, 16, 17, 18], 'cost': 180}, {'id': 46, 'elements': [11, 12], 'cost': 68}, {'id': 47, 'elements': [4, 6], 'cost': 40}, {'id': 48, 'elements': [10, 11, 12, 13], 'cost': 24}, {'id': 49, 'elements': [6, 7, 8, 9, 10], 'cost': 400}, {'id': 50, 'elements': [8, 9, 10, 12], 'cost': 16}, {'id': 51, 'elements': [5, 7, 9, 10], 'cost': 244}, {'id': 52, 'elements': [14, 16, 17], 'cost': 87}, {'id': 53, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 66, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 67, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 68, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 69, 'elements': [17], 'cost': 10000, 'singleton': True}, {'id': 70, 'elements': [18], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0000_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0000_bag_stats.png'}","[17, 37, 45, 51]",728.0,"{'num_elements': 18, 'num_sets': 70, 'sets': [{'id': 'S1', 'elements': [7, 8, 9, 11], 'cost': 48}, {'id': 'S2', 'elements': [1, 2, 3, 4], 'cost': 104}, {'id': 'S3', 'elements': [15, 16, 17, 18], 'cost': 392}, {'id': 'S4', 'elements': [11, 12, 13, 14, 17], 'cost': 65}, {'id': 'S5', 'elements': [12, 13, 14], 'cost': 207}, {'id': 'S6', 'elements': [10, 12], 'cost': 162}, {'id': 'S7', 'elements': [9, 10, 12, 13], 'cost': 152}, {'id': 'S8', 'elements': [5, 7, 8, 10, 11], 'cost': 230}, {'id': 'S9', 'elements': [6, 9, 10], 'cost': 246}, {'id': 'S10', 'elements': [6, 8], 'cost': 164}, {'id': 'S11', 'elements': [6, 8, 9, 10], 'cost': 20}, {'id': 'S12', 'elements': [11, 12, 13, 15, 16], 'cost': 420}, {'id': 'S13', 'elements': [13, 14, 15, 17], 'cost': 384}, {'id': 'S14', 'elements': [11, 12, 14, 18], 'cost': 388}, {'id': 'S15', 'elements': [2, 3], 'cost': 176}, {'id': 'S16', 'elements': [2, 3, 4], 'cost': 99}, {'id': 'S17', 'elements': [1, 2, 3, 4, 6], 'cost': 220}, {'id': 'S18', 'elements': [9, 12], 'cost': 186}, {'id': 'S19', 'elements': [6, 8, 9, 10, 12], 'cost': 410}, {'id': 'S20', 'elements': [5, 7, 8, 9], 'cost': 272}, {'id': 'S21', 'elements': [9, 10, 11, 13, 15], 'cost': 155}, {'id': 'S22', 'elements': [1, 3], 'cost': 18}, {'id': 'S23', 'elements': [14, 15, 16, 17, 18], 'cost': 390}, {'id': 'S24', 'elements': [7, 8, 10], 'cost': 276}, {'id': 'S25', 'elements': [9, 12, 13], 'cost': 87}, {'id': 'S26', 'elements': [1, 2], 'cost': 58}, {'id': 'S27', 'elements': [1, 5], 'cost': 86}, {'id': 'S28', 'elements': [7, 8], 'cost': 140}, {'id': 'S29', 'elements': [15, 16], 'cost': 106}, {'id': 'S30', 'elements': [3, 5], 'cost': 106}, {'id': 'S31', 'elements': [1, 3, 6], 'cost': 156}, {'id': 'S32', 'elements': [2, 3, 4, 5], 'cost': 144}, {'id': 'S33', 'elements': [3, 4, 5], 'cost': 21}, {'id': 'S34', 'elements': [16, 18], 'cost': 16}, {'id': 'S35', 'elements': [13, 14, 15, 16, 17], 'cost': 315}, {'id': 'S36', 'elements': [7, 9], 'cost': 190}, {'id': 'S37', 'elements': [8, 11, 12, 14], 'cost': 84}, {'id': 'S38', 'elements': [14, 15], 'cost': 154}, {'id': 'S39', 'elements': [8, 11], 'cost': 146}, {'id': 'S40', 'elements': [7, 8, 9, 10], 'cost': 344}, {'id': 'S41', 'elements': [12, 14, 15, 16], 'cost': 292}, {'id': 'S42', 'elements': [1, 2, 3, 4, 5], 'cost': 160}, {'id': 'S43', 'elements': [4, 6, 7], 'cost': 252}, {'id': 'S44', 'elements': [14, 16, 17, 18], 'cost': 60}, {'id': 'S45', 'elements': [13, 15, 16, 17, 18], 'cost': 180}, {'id': 'S46', 'elements': [11, 12], 'cost': 68}, {'id': 'S47', 'elements': [4, 6], 'cost': 40}, {'id': 'S48', 'elements': [10, 11, 12, 13], 'cost': 24}, {'id': 'S49', 'elements': [6, 7, 8, 9, 10], 'cost': 400}, {'id': 'S50', 'elements': [8, 9, 10, 12], 'cost': 16}, {'id': 'S51', 'elements': [5, 7, 9, 10], 'cost': 244}, {'id': 'S52', 'elements': [14, 16, 17], 'cost': 87}, {'id': 'S53', 'elements': [1], 'cost': 10000}, {'id': 'S54', 'elements': [2], 'cost': 10000}, {'id': 'S55', 'elements': [3], 'cost': 10000}, {'id': 'S56', 'elements': [4], 'cost': 10000}, {'id': 'S57', 'elements': [5], 'cost': 10000}, {'id': 'S58', 'elements': [6], 'cost': 10000}, {'id': 'S59', 'elements': [7], 'cost': 10000}, {'id': 'S60', 'elements': [8], 'cost': 10000}, {'id': 'S61', 'elements': [9], 'cost': 10000}, {'id': 'S62', 'elements': [10], 'cost': 10000}, {'id': 'S63', 'elements': [11], 'cost': 10000}, {'id': 'S64', 'elements': [12], 'cost': 10000}, {'id': 'S65', 'elements': [13], 'cost': 10000}, {'id': 'S66', 'elements': [14], 'cost': 10000}, {'id': 'S67', 'elements': [15], 'cost': 10000}, {'id': 'S68', 'elements': [16], 'cost': 10000}, {'id': 'S69', 'elements': [17], 'cost': 10000}, {'id': 'S70', 'elements': [18], 'cost': 10000}]}","['S17', 'S37', 'S45', 'S51']",1,csv,1
SPP,SPP,"Someone’s put together a list of possible buffet stations for an event, and the practical question is which of those stations to run so every dish on the menu gets served by one—and only one—station (no omissions, no duplicates). The measure of a better plan is simple: the lower the combined cost of the stations you pick, the better — just add their costs to get the total. The full details about dishes, stations, and prices appear below.

- **num_dishes**: 13
- **num_stations**: 54

| station_id | station_cost | station_dishes |
|---|---|---|
| S1 | 72 | F G |
| S2 | 138 | I L |
| S3 | 46 | A D |
| S4 | 123 | C D E |
| S5 | 50 | G K |
| S6 | 138 | K L |
| S7 | 258 | J K L |
| S8 | 160 | H I J K |
| S9 | 162 | D E F |
| S10 | 117 | G H I |
| S11 | 150 | A B |
| S12 | 255 | K L M |
| S13 | 120 | D E H |
| S14 | 122 | B C |
| S15 | 198 | I M |
| S16 | 28 | J K L M |
| S17 | 216 | B C D |
| S18 | 174 | G I K |
| S19 | 200 | C E |
| S20 | 36 | E F |
| S21 | 18 | G H |
| S22 | 184 | A B C D |
| S23 | 108 | H I |
| S24 | 87 | J L M |
| S25 | 174 | F H |
| S26 | 33 | A B C |
| S27 | 98 | G I |
| S28 | 45 | G |
| S29 | 51 | A C D |
| S30 | 297 | A B D |
| S31 | 30 | J K |
| S32 | 120 | A B E |
| S33 | 267 | E G H |
| S34 | 231 | B E F |
| S35 | 4 | D G |
| S36 | 364 | F H I J |
| S37 | 70 | D E |
| S38 | 228 | I J K |
| S39 | 116 | B E |
| S40 | 154 | L M |
| S41 | 93 | C E F |
| S42 | 10000 | A |
| S43 | 10000 | B |
| S44 | 10000 | C |
| S45 | 10000 | D |
| S46 | 10000 | E |
| S47 | 10000 | F |
| S48 | 10000 | G |
| S49 | 10000 | H |
| S50 | 10000 | I |
| S51 | 10000 | J |
| S52 | 10000 | K |
| S53 | 10000 | L |
| S54 | 10000 | M |

Also, if you're sending back the chosen stations, a neat little JSON snippet like this makes it easy to check:

{
  ""solution"": [""station_id"", ...]
}

Here ""solution"" is just the list of station IDs you plan to run — one entry per station. The placeholder ""station_id"" is where you'd drop the actual identifier for a buffet station from the instance (so replace that placeholder with the real ID). This is only a sketch of the expected shape, not the actual answer.

Please make sure you use the exact identifiers from the instance input — don't rename them or invent new labels. Valid identifiers look like plain numbers such as ""1"" or ""23"", single capital letters like ""A"" or ""B"", or a capital letter followed by digits like ""A1"" or ""X7"".","{'problem_type': 'SPP', 'num_elements': 13, 'num_sets': 54, 'density': 0.16951566951566951, 'sets': [{'id': 1, 'elements': [6, 7], 'cost': 72}, {'id': 2, 'elements': [9, 12], 'cost': 138}, {'id': 3, 'elements': [1, 4], 'cost': 46}, {'id': 4, 'elements': [3, 4, 5], 'cost': 123}, {'id': 5, 'elements': [7, 11], 'cost': 50}, {'id': 6, 'elements': [11, 12], 'cost': 138}, {'id': 7, 'elements': [10, 11, 12], 'cost': 258}, {'id': 8, 'elements': [8, 9, 10, 11], 'cost': 160}, {'id': 9, 'elements': [4, 5, 6], 'cost': 162}, {'id': 10, 'elements': [7, 8, 9], 'cost': 117}, {'id': 11, 'elements': [1, 2], 'cost': 150}, {'id': 12, 'elements': [11, 12, 13], 'cost': 255}, {'id': 13, 'elements': [4, 5, 8], 'cost': 120}, {'id': 14, 'elements': [2, 3], 'cost': 122}, {'id': 15, 'elements': [9, 13], 'cost': 198}, {'id': 16, 'elements': [10, 11, 12, 13], 'cost': 28}, {'id': 17, 'elements': [2, 3, 4], 'cost': 216}, {'id': 18, 'elements': [7, 9, 11], 'cost': 174}, {'id': 19, 'elements': [3, 5], 'cost': 200}, {'id': 20, 'elements': [5, 6], 'cost': 36}, {'id': 21, 'elements': [7, 8], 'cost': 18}, {'id': 22, 'elements': [1, 2, 3, 4], 'cost': 184}, {'id': 23, 'elements': [8, 9], 'cost': 108}, {'id': 24, 'elements': [10, 12, 13], 'cost': 87}, {'id': 25, 'elements': [6, 8], 'cost': 174}, {'id': 26, 'elements': [1, 2, 3], 'cost': 33}, {'id': 27, 'elements': [7, 9], 'cost': 98}, {'id': 28, 'elements': [7], 'cost': 45}, {'id': 29, 'elements': [1, 3, 4], 'cost': 51}, {'id': 30, 'elements': [1, 2, 4], 'cost': 297}, {'id': 31, 'elements': [10, 11], 'cost': 30}, {'id': 32, 'elements': [1, 2, 5], 'cost': 120}, {'id': 33, 'elements': [5, 7, 8], 'cost': 267}, {'id': 34, 'elements': [2, 5, 6], 'cost': 231}, {'id': 35, 'elements': [4, 7], 'cost': 4}, {'id': 36, 'elements': [6, 8, 9, 10], 'cost': 364}, {'id': 37, 'elements': [4, 5], 'cost': 70}, {'id': 38, 'elements': [9, 10, 11], 'cost': 228}, {'id': 39, 'elements': [2, 5], 'cost': 116}, {'id': 40, 'elements': [12, 13], 'cost': 154}, {'id': 41, 'elements': [3, 5, 6], 'cost': 93}, {'id': 42, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 43, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 44, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 45, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 46, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 47, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [13], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0001_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0001_bag_stats.png'}","[16, 20, 23, 26, 35]",209.0,"{'num_elements': 13, 'num_sets': 54, 'sets': [{'id': 'S1', 'elements': ['F', 'G'], 'cost': 72}, {'id': 'S2', 'elements': ['I', 'L'], 'cost': 138}, {'id': 'S3', 'elements': ['A', 'D'], 'cost': 46}, {'id': 'S4', 'elements': ['C', 'D', 'E'], 'cost': 123}, {'id': 'S5', 'elements': ['G', 'K'], 'cost': 50}, {'id': 'S6', 'elements': ['K', 'L'], 'cost': 138}, {'id': 'S7', 'elements': ['J', 'K', 'L'], 'cost': 258}, {'id': 'S8', 'elements': ['H', 'I', 'J', 'K'], 'cost': 160}, {'id': 'S9', 'elements': ['D', 'E', 'F'], 'cost': 162}, {'id': 'S10', 'elements': ['G', 'H', 'I'], 'cost': 117}, {'id': 'S11', 'elements': ['A', 'B'], 'cost': 150}, {'id': 'S12', 'elements': ['K', 'L', 'M'], 'cost': 255}, {'id': 'S13', 'elements': ['D', 'E', 'H'], 'cost': 120}, {'id': 'S14', 'elements': ['B', 'C'], 'cost': 122}, {'id': 'S15', 'elements': ['I', 'M'], 'cost': 198}, {'id': 'S16', 'elements': ['J', 'K', 'L', 'M'], 'cost': 28}, {'id': 'S17', 'elements': ['B', 'C', 'D'], 'cost': 216}, {'id': 'S18', 'elements': ['G', 'I', 'K'], 'cost': 174}, {'id': 'S19', 'elements': ['C', 'E'], 'cost': 200}, {'id': 'S20', 'elements': ['E', 'F'], 'cost': 36}, {'id': 'S21', 'elements': ['G', 'H'], 'cost': 18}, {'id': 'S22', 'elements': ['A', 'B', 'C', 'D'], 'cost': 184}, {'id': 'S23', 'elements': ['H', 'I'], 'cost': 108}, {'id': 'S24', 'elements': ['J', 'L', 'M'], 'cost': 87}, {'id': 'S25', 'elements': ['F', 'H'], 'cost': 174}, {'id': 'S26', 'elements': ['A', 'B', 'C'], 'cost': 33}, {'id': 'S27', 'elements': ['G', 'I'], 'cost': 98}, {'id': 'S28', 'elements': ['G'], 'cost': 45}, {'id': 'S29', 'elements': ['A', 'C', 'D'], 'cost': 51}, {'id': 'S30', 'elements': ['A', 'B', 'D'], 'cost': 297}, {'id': 'S31', 'elements': ['J', 'K'], 'cost': 30}, {'id': 'S32', 'elements': ['A', 'B', 'E'], 'cost': 120}, {'id': 'S33', 'elements': ['E', 'G', 'H'], 'cost': 267}, {'id': 'S34', 'elements': ['B', 'E', 'F'], 'cost': 231}, {'id': 'S35', 'elements': ['D', 'G'], 'cost': 4}, {'id': 'S36', 'elements': ['F', 'H', 'I', 'J'], 'cost': 364}, {'id': 'S37', 'elements': ['D', 'E'], 'cost': 70}, {'id': 'S38', 'elements': ['I', 'J', 'K'], 'cost': 228}, {'id': 'S39', 'elements': ['B', 'E'], 'cost': 116}, {'id': 'S40', 'elements': ['L', 'M'], 'cost': 154}, {'id': 'S41', 'elements': ['C', 'E', 'F'], 'cost': 93}, {'id': 'S42', 'elements': ['A'], 'cost': 10000}, {'id': 'S43', 'elements': ['B'], 'cost': 10000}, {'id': 'S44', 'elements': ['C'], 'cost': 10000}, {'id': 'S45', 'elements': ['D'], 'cost': 10000}, {'id': 'S46', 'elements': ['E'], 'cost': 10000}, {'id': 'S47', 'elements': ['F'], 'cost': 10000}, {'id': 'S48', 'elements': ['G'], 'cost': 10000}, {'id': 'S49', 'elements': ['H'], 'cost': 10000}, {'id': 'S50', 'elements': ['I'], 'cost': 10000}, {'id': 'S51', 'elements': ['J'], 'cost': 10000}, {'id': 'S52', 'elements': ['K'], 'cost': 10000}, {'id': 'S53', 'elements': ['L'], 'cost': 10000}, {'id': 'S54', 'elements': ['M'], 'cost': 10000}]}","['S16', 'S20', 'S23', 'S26', 'S35']",2,markdown_table,names
SPP,SPP,"We’ve got a stack of classes that need spaces, and the job is to choose rooms so each class is assigned to one room only (no class in two rooms, no class left without a room). Every room that’s used carries a rental fee, and the total cost is just the sum of those room fees — the goal is to make that sum as small as it can be. The exact rooms, class lists and prices follow below.

- **num_classes**: 18
- **num_rooms**: 81

| room_id | rental_cost | classes_in_room |
|---|---|---|
| S1 | 355 | A B C D E |
| S2 | 160 | M N O R |
| S3 | 294 | C E F |
| S4 | 272 | F G I K |
| S5 | 364 | B C D F |
| S6 | 328 | F G I L |
| S7 | 35 | L N O P Q |
| S8 | 222 | M N P |
| S9 | 128 | A B D E |
| S10 | 58 | N P |
| S11 | 400 | K L M O R |
| S12 | 56 | C G |
| S13 | 3 | A B C |
| S14 | 270 | N O Q |
| S15 | 130 | B C D E F |
| S16 | 57 | O P Q |
| S17 | 146 | F G |
| S18 | 320 | M N O P R |
| S19 | 12 | B C D |
| S20 | 70 | B G |
| S21 | 300 | D F G I |
| S22 | 126 | O P R |
| S23 | 392 | D E G I |
| S24 | 264 | L M N Q |
| S25 | 160 | A D E F |
| S26 | 64 | N P Q R |
| S27 | 248 | C D E F |
| S28 | 90 | F G H I J |
| S29 | 5 | F H I J K |
| S30 | 60 | J K L |
| S31 | 150 | J M O |
| S32 | 147 | K N O |
| S33 | 228 | B E F |
| S34 | 177 | C D F |
| S35 | 40 | E F |
| S36 | 168 | H J |
| S37 | 98 | L M |
| S38 | 150 | M N P Q R |
| S39 | 88 | K N O Q |
| S40 | 76 | A B C D |
| S41 | 33 | L N P |
| S42 | 240 | K L N P |
| S43 | 348 | K N P Q |
| S44 | 220 | B C D E |
| S45 | 124 | M N O Q |
| S46 | 255 | K L P |
| S47 | 200 | Q R |
| S48 | 440 | J K L M N |
| S49 | 280 | A B C E |
| S50 | 310 | J L M N O |
| S51 | 145 | G H I K N |
| S52 | 345 | I L N O P |
| S53 | 288 | H J K O |
| S54 | 270 | G H I |
| S55 | 192 | K L M |
| S56 | 118 | F J |
| S57 | 162 | O Q R |
| S58 | 264 | L P R |
| S59 | 177 | I K P |
| S60 | 6 | G H |
| S61 | 146 | I J |
| S62 | 129 | P Q R |
| S63 | 75 | A C D E F |
| S64 | 10000 | A |
| S65 | 10000 | B |
| S66 | 10000 | C |
| S67 | 10000 | D |
| S68 | 10000 | E |
| S69 | 10000 | F |
| S70 | 10000 | G |
| S71 | 10000 | H |
| S72 | 10000 | I |
| S73 | 10000 | J |
| S74 | 10000 | K |
| S75 | 10000 | L |
| S76 | 10000 | M |
| S77 | 10000 | N |
| S78 | 10000 | O |
| S79 | 10000 | P |
| S80 | 10000 | Q |
| S81 | 10000 | R |

When you tell me which rooms you've picked, just drop them into this simple JSON shape so it's easy to parse:

{
  ""solution"": [""room_id"", ...]
}

Just to keep it friendly: ""solution"" is the list of rooms you're going to rent (one entry per chosen room). Each ""room_id"" in that list should be the exact identifier from the input — it's just a placeholder here to show the shape, not the actual answer.

Please use the identifiers exactly as they appear in the instance input — do not rename them or invent new labels. For example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 18, 'num_sets': 81, 'density': 0.16529492455418382, 'sets': [{'id': 1, 'elements': [1, 2, 3, 4, 5], 'cost': 355}, {'id': 2, 'elements': [13, 14, 15, 18], 'cost': 160}, {'id': 3, 'elements': [3, 5, 6], 'cost': 294}, {'id': 4, 'elements': [6, 7, 9, 11], 'cost': 272}, {'id': 5, 'elements': [2, 3, 4, 6], 'cost': 364}, {'id': 6, 'elements': [6, 7, 9, 12], 'cost': 328}, {'id': 7, 'elements': [12, 14, 15, 16, 17], 'cost': 35}, {'id': 8, 'elements': [13, 14, 16], 'cost': 222}, {'id': 9, 'elements': [1, 2, 4, 5], 'cost': 128}, {'id': 10, 'elements': [14, 16], 'cost': 58}, {'id': 11, 'elements': [11, 12, 13, 15, 18], 'cost': 400}, {'id': 12, 'elements': [3, 7], 'cost': 56}, {'id': 13, 'elements': [1, 2, 3], 'cost': 3}, {'id': 14, 'elements': [14, 15, 17], 'cost': 270}, {'id': 15, 'elements': [2, 3, 4, 5, 6], 'cost': 130}, {'id': 16, 'elements': [15, 16, 17], 'cost': 57}, {'id': 17, 'elements': [6, 7], 'cost': 146}, {'id': 18, 'elements': [13, 14, 15, 16, 18], 'cost': 320}, {'id': 19, 'elements': [2, 3, 4], 'cost': 12}, {'id': 20, 'elements': [2, 7], 'cost': 70}, {'id': 21, 'elements': [4, 6, 7, 9], 'cost': 300}, {'id': 22, 'elements': [15, 16, 18], 'cost': 126}, {'id': 23, 'elements': [4, 5, 7, 9], 'cost': 392}, {'id': 24, 'elements': [12, 13, 14, 17], 'cost': 264}, {'id': 25, 'elements': [1, 4, 5, 6], 'cost': 160}, {'id': 26, 'elements': [14, 16, 17, 18], 'cost': 64}, {'id': 27, 'elements': [3, 4, 5, 6], 'cost': 248}, {'id': 28, 'elements': [6, 7, 8, 9, 10], 'cost': 90}, {'id': 29, 'elements': [6, 8, 9, 10, 11], 'cost': 5}, {'id': 30, 'elements': [10, 11, 12], 'cost': 60}, {'id': 31, 'elements': [10, 13, 15], 'cost': 150}, {'id': 32, 'elements': [11, 14, 15], 'cost': 147}, {'id': 33, 'elements': [2, 5, 6], 'cost': 228}, {'id': 34, 'elements': [3, 4, 6], 'cost': 177}, {'id': 35, 'elements': [5, 6], 'cost': 40}, {'id': 36, 'elements': [8, 10], 'cost': 168}, {'id': 37, 'elements': [12, 13], 'cost': 98}, {'id': 38, 'elements': [13, 14, 16, 17, 18], 'cost': 150}, {'id': 39, 'elements': [11, 14, 15, 17], 'cost': 88}, {'id': 40, 'elements': [1, 2, 3, 4], 'cost': 76}, {'id': 41, 'elements': [12, 14, 16], 'cost': 33}, {'id': 42, 'elements': [11, 12, 14, 16], 'cost': 240}, {'id': 43, 'elements': [11, 14, 16, 17], 'cost': 348}, {'id': 44, 'elements': [2, 3, 4, 5], 'cost': 220}, {'id': 45, 'elements': [13, 14, 15, 17], 'cost': 124}, {'id': 46, 'elements': [11, 12, 16], 'cost': 255}, {'id': 47, 'elements': [17, 18], 'cost': 200}, {'id': 48, 'elements': [10, 11, 12, 13, 14], 'cost': 440}, {'id': 49, 'elements': [1, 2, 3, 5], 'cost': 280}, {'id': 50, 'elements': [10, 12, 13, 14, 15], 'cost': 310}, {'id': 51, 'elements': [7, 8, 9, 11, 14], 'cost': 145}, {'id': 52, 'elements': [9, 12, 14, 15, 16], 'cost': 345}, {'id': 53, 'elements': [8, 10, 11, 15], 'cost': 288}, {'id': 54, 'elements': [7, 8, 9], 'cost': 270}, {'id': 55, 'elements': [11, 12, 13], 'cost': 192}, {'id': 56, 'elements': [6, 10], 'cost': 118}, {'id': 57, 'elements': [15, 17, 18], 'cost': 162}, {'id': 58, 'elements': [12, 16, 18], 'cost': 264}, {'id': 59, 'elements': [9, 11, 16], 'cost': 177}, {'id': 60, 'elements': [7, 8], 'cost': 6}, {'id': 61, 'elements': [9, 10], 'cost': 146}, {'id': 62, 'elements': [16, 17, 18], 'cost': 129}, {'id': 63, 'elements': [1, 3, 4, 5, 6], 'cost': 75}, {'id': 64, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 66, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 67, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 68, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 69, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 70, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 71, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 72, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 73, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 74, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 75, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 76, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 77, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 78, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 79, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 80, 'elements': [17], 'cost': 10000, 'singleton': True}, {'id': 81, 'elements': [18], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0002_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0002_bag_stats.png'}","[9, 10, 12, 29, 37, 57]",507.0,"{'num_elements': 18, 'num_sets': 81, 'sets': [{'id': 'S1', 'elements': ['A', 'B', 'C', 'D', 'E'], 'cost': 355}, {'id': 'S2', 'elements': ['M', 'N', 'O', 'R'], 'cost': 160}, {'id': 'S3', 'elements': ['C', 'E', 'F'], 'cost': 294}, {'id': 'S4', 'elements': ['F', 'G', 'I', 'K'], 'cost': 272}, {'id': 'S5', 'elements': ['B', 'C', 'D', 'F'], 'cost': 364}, {'id': 'S6', 'elements': ['F', 'G', 'I', 'L'], 'cost': 328}, {'id': 'S7', 'elements': ['L', 'N', 'O', 'P', 'Q'], 'cost': 35}, {'id': 'S8', 'elements': ['M', 'N', 'P'], 'cost': 222}, {'id': 'S9', 'elements': ['A', 'B', 'D', 'E'], 'cost': 128}, {'id': 'S10', 'elements': ['N', 'P'], 'cost': 58}, {'id': 'S11', 'elements': ['K', 'L', 'M', 'O', 'R'], 'cost': 400}, {'id': 'S12', 'elements': ['C', 'G'], 'cost': 56}, {'id': 'S13', 'elements': ['A', 'B', 'C'], 'cost': 3}, {'id': 'S14', 'elements': ['N', 'O', 'Q'], 'cost': 270}, {'id': 'S15', 'elements': ['B', 'C', 'D', 'E', 'F'], 'cost': 130}, {'id': 'S16', 'elements': ['O', 'P', 'Q'], 'cost': 57}, {'id': 'S17', 'elements': ['F', 'G'], 'cost': 146}, {'id': 'S18', 'elements': ['M', 'N', 'O', 'P', 'R'], 'cost': 320}, {'id': 'S19', 'elements': ['B', 'C', 'D'], 'cost': 12}, {'id': 'S20', 'elements': ['B', 'G'], 'cost': 70}, {'id': 'S21', 'elements': ['D', 'F', 'G', 'I'], 'cost': 300}, {'id': 'S22', 'elements': ['O', 'P', 'R'], 'cost': 126}, {'id': 'S23', 'elements': ['D', 'E', 'G', 'I'], 'cost': 392}, {'id': 'S24', 'elements': ['L', 'M', 'N', 'Q'], 'cost': 264}, {'id': 'S25', 'elements': ['A', 'D', 'E', 'F'], 'cost': 160}, {'id': 'S26', 'elements': ['N', 'P', 'Q', 'R'], 'cost': 64}, {'id': 'S27', 'elements': ['C', 'D', 'E', 'F'], 'cost': 248}, {'id': 'S28', 'elements': ['F', 'G', 'H', 'I', 'J'], 'cost': 90}, {'id': 'S29', 'elements': ['F', 'H', 'I', 'J', 'K'], 'cost': 5}, {'id': 'S30', 'elements': ['J', 'K', 'L'], 'cost': 60}, {'id': 'S31', 'elements': ['J', 'M', 'O'], 'cost': 150}, {'id': 'S32', 'elements': ['K', 'N', 'O'], 'cost': 147}, {'id': 'S33', 'elements': ['B', 'E', 'F'], 'cost': 228}, {'id': 'S34', 'elements': ['C', 'D', 'F'], 'cost': 177}, {'id': 'S35', 'elements': ['E', 'F'], 'cost': 40}, {'id': 'S36', 'elements': ['H', 'J'], 'cost': 168}, {'id': 'S37', 'elements': ['L', 'M'], 'cost': 98}, {'id': 'S38', 'elements': ['M', 'N', 'P', 'Q', 'R'], 'cost': 150}, {'id': 'S39', 'elements': ['K', 'N', 'O', 'Q'], 'cost': 88}, {'id': 'S40', 'elements': ['A', 'B', 'C', 'D'], 'cost': 76}, {'id': 'S41', 'elements': ['L', 'N', 'P'], 'cost': 33}, {'id': 'S42', 'elements': ['K', 'L', 'N', 'P'], 'cost': 240}, {'id': 'S43', 'elements': ['K', 'N', 'P', 'Q'], 'cost': 348}, {'id': 'S44', 'elements': ['B', 'C', 'D', 'E'], 'cost': 220}, {'id': 'S45', 'elements': ['M', 'N', 'O', 'Q'], 'cost': 124}, {'id': 'S46', 'elements': ['K', 'L', 'P'], 'cost': 255}, {'id': 'S47', 'elements': ['Q', 'R'], 'cost': 200}, {'id': 'S48', 'elements': ['J', 'K', 'L', 'M', 'N'], 'cost': 440}, {'id': 'S49', 'elements': ['A', 'B', 'C', 'E'], 'cost': 280}, {'id': 'S50', 'elements': ['J', 'L', 'M', 'N', 'O'], 'cost': 310}, {'id': 'S51', 'elements': ['G', 'H', 'I', 'K', 'N'], 'cost': 145}, {'id': 'S52', 'elements': ['I', 'L', 'N', 'O', 'P'], 'cost': 345}, {'id': 'S53', 'elements': ['H', 'J', 'K', 'O'], 'cost': 288}, {'id': 'S54', 'elements': ['G', 'H', 'I'], 'cost': 270}, {'id': 'S55', 'elements': ['K', 'L', 'M'], 'cost': 192}, {'id': 'S56', 'elements': ['F', 'J'], 'cost': 118}, {'id': 'S57', 'elements': ['O', 'Q', 'R'], 'cost': 162}, {'id': 'S58', 'elements': ['L', 'P', 'R'], 'cost': 264}, {'id': 'S59', 'elements': ['I', 'K', 'P'], 'cost': 177}, {'id': 'S60', 'elements': ['G', 'H'], 'cost': 6}, {'id': 'S61', 'elements': ['I', 'J'], 'cost': 146}, {'id': 'S62', 'elements': ['P', 'Q', 'R'], 'cost': 129}, {'id': 'S63', 'elements': ['A', 'C', 'D', 'E', 'F'], 'cost': 75}, {'id': 'S64', 'elements': ['A'], 'cost': 10000}, {'id': 'S65', 'elements': ['B'], 'cost': 10000}, {'id': 'S66', 'elements': ['C'], 'cost': 10000}, {'id': 'S67', 'elements': ['D'], 'cost': 10000}, {'id': 'S68', 'elements': ['E'], 'cost': 10000}, {'id': 'S69', 'elements': ['F'], 'cost': 10000}, {'id': 'S70', 'elements': ['G'], 'cost': 10000}, {'id': 'S71', 'elements': ['H'], 'cost': 10000}, {'id': 'S72', 'elements': ['I'], 'cost': 10000}, {'id': 'S73', 'elements': ['J'], 'cost': 10000}, {'id': 'S74', 'elements': ['K'], 'cost': 10000}, {'id': 'S75', 'elements': ['L'], 'cost': 10000}, {'id': 'S76', 'elements': ['M'], 'cost': 10000}, {'id': 'S77', 'elements': ['N'], 'cost': 10000}, {'id': 'S78', 'elements': ['O'], 'cost': 10000}, {'id': 'S79', 'elements': ['P'], 'cost': 10000}, {'id': 'S80', 'elements': ['Q'], 'cost': 10000}, {'id': 'S81', 'elements': ['R'], 'cost': 10000}]}","['S9', 'S10', 'S12', 'S29', 'S37', 'S57']",3,markdown_table,names
SPP,SPP,"We’re sorting out which warehouse zones to power up so each item gets stored in exactly one chosen zone and the total cost stays as low as it can be — simply add up the activation cost of each zone we pick. Nothing can be left unassigned or placed in two spots. Concrete details about items, zones and prices follow below.

{
  ""total_products"": 17,
  ""candidate_zones_count"": 68,
  ""sets"": [
    {
      ""zone_id"": ""S1"",
      ""products"": [
        13,
        15
      ],
      ""activation_cost"": 28
    },
    {
      ""zone_id"": ""S2"",
      ""products"": [
        4,
        8,
        10
      ],
      ""activation_cost"": 255
    },
    {
      ""zone_id"": ""S3"",
      ""products"": [
        11,
        14,
        16
      ],
      ""activation_cost"": 156
    },
    {
      ""zone_id"": ""S4"",
      ""products"": [
        1,
        3,
        6
      ],
      ""activation_cost"": 33
    },
    {
      ""zone_id"": ""S5"",
      ""products"": [
        2,
        3,
        5
      ],
      ""activation_cost"": 213
    },
    {
      ""zone_id"": ""S6"",
      ""products"": [
        11,
        12,
        13,
        16
      ],
      ""activation_cost"": 52
    },
    {
      ""zone_id"": ""S7"",
      ""products"": [
        2,
        3,
        4,
        5
      ],
      ""activation_cost"": 112
    },
    {
      ""zone_id"": ""S8"",
      ""products"": [
        6,
        7,
        9
      ],
      ""activation_cost"": 282
    },
    {
      ""zone_id"": ""S9"",
      ""products"": [
        12,
        13
      ],
      ""activation_cost"": 40
    },
    {
      ""zone_id"": ""S10"",
      ""products"": [
        12,
        13,
        14,
        16
      ],
      ""activation_cost"": 380
    },
    {
      ""zone_id"": ""S11"",
      ""products"": [
        3,
        4,
        5
      ],
      ""activation_cost"": 129
    },
    {
      ""zone_id"": ""S12"",
      ""products"": [
        12,
        13,
        14,
        15
      ],
      ""activation_cost"": 372
    },
    {
      ""zone_id"": ""S13"",
      ""products"": [
        8,
        9
      ],
      ""activation_cost"": 148
    },
    {
      ""zone_id"": ""S14"",
      ""products"": [
        9,
        10,
        11,
        12,
        13
      ],
      ""activation_cost"": 495
    },
    {
      ""zone_id"": ""S15"",
      ""products"": [
        0,
        1,
        2,
        3
      ],
      ""activation_cost"": 140
    },
    {
      ""zone_id"": ""S16"",
      ""products"": [
        7,
        8,
        9,
        11
      ],
      ""activation_cost"": 340
    },
    {
      ""zone_id"": ""S17"",
      ""products"": [
        2,
        5,
        7
      ],
      ""activation_cost"": 204
    },
    {
      ""zone_id"": ""S18"",
      ""products"": [
        11,
        13
      ],
      ""activation_cost"": 80
    },
    {
      ""zone_id"": ""S19"",
      ""products"": [
        0,
        1,
        3,
        4
      ],
      ""activation_cost"": 44
    },
    {
      ""zone_id"": ""S20"",
      ""products"": [
        11,
        12,
        13
      ],
      ""activation_cost"": 291
    },
    {
      ""zone_id"": ""S21"",
      ""products"": [
        10,
        11,
        13
      ],
      ""activation_cost"": 189
    },
    {
      ""zone_id"": ""S22"",
      ""products"": [
        6,
        9,
        10
      ],
      ""activation_cost"": 3
    },
    {
      ""zone_id"": ""S23"",
      ""products"": [
        6,
        7
      ],
      ""activation_cost"": 76
    },
    {
      ""zone_id"": ""S24"",
      ""products"": [
        0,
        1,
        2
      ],
      ""activation_cost"": 207
    },
    {
      ""zone_id"": ""S25"",
      ""products"": [
        13,
        14,
        15,
        16
      ],
      ""activation_cost"": 380
    },
    {
      ""zone_id"": ""S26"",
      ""products"": [
        0,
        1
      ],
      ""activation_cost"": 82
    },
    {
      ""zone_id"": ""S27"",
      ""products"": [
        8,
        10,
        11,
        13
      ],
      ""activation_cost"": 168
    },
    {
      ""zone_id"": ""S28"",
      ""products"": [
        15,
        16
      ],
      ""activation_cost"": 24
    },
    {
      ""zone_id"": ""S29"",
      ""products"": [
        2,
        3,
        4
      ],
      ""activation_cost"": 153
    },
    {
      ""zone_id"": ""S30"",
      ""products"": [
        7,
        9,
        10,
        11,
        12
      ],
      ""activation_cost"": 125
    },
    {
      ""zone_id"": ""S31"",
      ""products"": [
        8,
        9,
        10,
        12,
        14
      ],
      ""activation_cost"": 165
    },
    {
      ""zone_id"": ""S32"",
      ""products"": [
        4,
        5
      ],
      ""activation_cost"": 46
    },
    {
      ""zone_id"": ""S33"",
      ""products"": [
        12,
        13,
        14,
        15,
        16
      ],
      ""activation_cost"": 415
    },
    {
      ""zone_id"": ""S34"",
      ""products"": [
        7,
        8,
        10,
        12
      ],
      ""activation_cost"": 228
    },
    {
      ""zone_id"": ""S35"",
      ""products"": [
        5,
        7,
        8,
        9
      ],
      ""activation_cost"": 20
    },
    {
      ""zone_id"": ""S36"",
      ""products"": [
        7,
        11
      ],
      ""activation_cost"": 24
    },
    {
      ""zone_id"": ""S37"",
      ""products"": [
        5,
        6
      ],
      ""activation_cost"": 40
    },
    {
      ""zone_id"": ""S38"",
      ""products"": [
        8,
        11
      ],
      ""activation_cost"": 136
    },
    {
      ""zone_id"": ""S39"",
      ""products"": [
        4,
        5,
        8
      ],
      ""activation_cost"": 300
    },
    {
      ""zone_id"": ""S40"",
      ""products"": [
        0,
        2,
        3
      ],
      ""activation_cost"": 39
    },
    {
      ""zone_id"": ""S41"",
      ""products"": [
        11,
        12,
        13,
        15
      ],
      ""activation_cost"": 80
    },
    {
      ""zone_id"": ""S42"",
      ""products"": [
        3,
        5,
        6
      ],
      ""activation_cost"": 255
    },
    {
      ""zone_id"": ""S43"",
      ""products"": [
        1,
        2,
        4
      ],
      ""activation_cost"": 51
    },
    {
      ""zone_id"": ""S44"",
      ""products"": [
        11,
        13,
        15,
        16
      ],
      ""activation_cost"": 160
    },
    {
      ""zone_id"": ""S45"",
      ""products"": [
        5,
        7
      ],
      ""activation_cost"": 50
    },
    {
      ""zone_id"": ""S46"",
      ""products"": [
        2,
        3
      ],
      ""activation_cost"": 26
    },
    {
      ""zone_id"": ""S47"",
      ""products"": [
        13,
        15,
        16
      ],
      ""activation_cost"": 51
    },
    {
      ""zone_id"": ""S48"",
      ""products"": [
        2,
        4,
        5,
        6
      ],
      ""activation_cost"": 88
    },
    {
      ""zone_id"": ""S49"",
      ""products"": [
        12,
        13,
        16
      ],
      ""activation_cost"": 183
    },
    {
      ""zone_id"": ""S50"",
      ""products"": [
        4,
        5,
        6,
        9,
        10
      ],
      ""activation_cost"": 245
    },
    {
      ""zone_id"": ""S51"",
      ""products"": [
        13,
        14,
        16
      ],
      ""activation_cost"": 54
    },
    {
      ""zone_id"": ""S52"",
      ""products"": [
        0
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S53"",
      ""products"": [
        1
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S54"",
      ""products"": [
        2
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S55"",
      ""products"": [
        3
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S56"",
      ""products"": [
        4
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S57"",
      ""products"": [
        5
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S58"",
      ""products"": [
        6
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S59"",
      ""products"": [
        7
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S60"",
      ""products"": [
        8
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S61"",
      ""products"": [
        9
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S62"",
      ""products"": [
        10
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S63"",
      ""products"": [
        11
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S64"",
      ""products"": [
        12
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S65"",
      ""products"": [
        13
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S66"",
      ""products"": [
        14
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S67"",
      ""products"": [
        15
      ],
      ""activation_cost"": 10000
    },
    {
      ""zone_id"": ""S68"",
      ""products"": [
        16
      ],
      ""activation_cost"": 10000
    }
  ]
}

Also, when you send back the chosen zones, do it in this relaxed little JSON shape — just a list of the zone identifiers you picked:

{
  ""solution"": [""zone_id"", ...]
}

""solution"" is the array where you put the ids of the zones to power up. The placeholder ""zone_id"" is just showing the expected form — replace it with the actual ids from the instance. This JSON is only a sketch of the shape I expect, not the final answer itself.

Please use the identifiers exactly as they appear in the instance input — no renaming, no new labels. 
- for example: ""Valid identifiers look like plain numbers such as ""1"" or ""23"", single capital letters like ""A"" or ""B"", or a capital letter followed by digits like ""A1"" or ""X7"".""","{'problem_type': 'SPP', 'num_elements': 17, 'num_sets': 68, 'density': 0.15657439446366783, 'sets': [{'id': 1, 'elements': [14, 16], 'cost': 28}, {'id': 2, 'elements': [5, 9, 11], 'cost': 255}, {'id': 3, 'elements': [12, 15, 17], 'cost': 156}, {'id': 4, 'elements': [2, 4, 7], 'cost': 33}, {'id': 5, 'elements': [3, 4, 6], 'cost': 213}, {'id': 6, 'elements': [12, 13, 14, 17], 'cost': 52}, {'id': 7, 'elements': [3, 4, 5, 6], 'cost': 112}, {'id': 8, 'elements': [7, 8, 10], 'cost': 282}, {'id': 9, 'elements': [13, 14], 'cost': 40}, {'id': 10, 'elements': [13, 14, 15, 17], 'cost': 380}, {'id': 11, 'elements': [4, 5, 6], 'cost': 129}, {'id': 12, 'elements': [13, 14, 15, 16], 'cost': 372}, {'id': 13, 'elements': [9, 10], 'cost': 148}, {'id': 14, 'elements': [10, 11, 12, 13, 14], 'cost': 495}, {'id': 15, 'elements': [1, 2, 3, 4], 'cost': 140}, {'id': 16, 'elements': [8, 9, 10, 12], 'cost': 340}, {'id': 17, 'elements': [3, 6, 8], 'cost': 204}, {'id': 18, 'elements': [12, 14], 'cost': 80}, {'id': 19, 'elements': [1, 2, 4, 5], 'cost': 44}, {'id': 20, 'elements': [12, 13, 14], 'cost': 291}, {'id': 21, 'elements': [11, 12, 14], 'cost': 189}, {'id': 22, 'elements': [7, 10, 11], 'cost': 3}, {'id': 23, 'elements': [7, 8], 'cost': 76}, {'id': 24, 'elements': [1, 2, 3], 'cost': 207}, {'id': 25, 'elements': [14, 15, 16, 17], 'cost': 380}, {'id': 26, 'elements': [1, 2], 'cost': 82}, {'id': 27, 'elements': [9, 11, 12, 14], 'cost': 168}, {'id': 28, 'elements': [16, 17], 'cost': 24}, {'id': 29, 'elements': [3, 4, 5], 'cost': 153}, {'id': 30, 'elements': [8, 10, 11, 12, 13], 'cost': 125}, {'id': 31, 'elements': [9, 10, 11, 13, 15], 'cost': 165}, {'id': 32, 'elements': [5, 6], 'cost': 46}, {'id': 33, 'elements': [13, 14, 15, 16, 17], 'cost': 415}, {'id': 34, 'elements': [8, 9, 11, 13], 'cost': 228}, {'id': 35, 'elements': [6, 8, 9, 10], 'cost': 20}, {'id': 36, 'elements': [8, 12], 'cost': 24}, {'id': 37, 'elements': [6, 7], 'cost': 40}, {'id': 38, 'elements': [9, 12], 'cost': 136}, {'id': 39, 'elements': [5, 6, 9], 'cost': 300}, {'id': 40, 'elements': [1, 3, 4], 'cost': 39}, {'id': 41, 'elements': [12, 13, 14, 16], 'cost': 80}, {'id': 42, 'elements': [4, 6, 7], 'cost': 255}, {'id': 43, 'elements': [2, 3, 5], 'cost': 51}, {'id': 44, 'elements': [12, 14, 16, 17], 'cost': 160}, {'id': 45, 'elements': [6, 8], 'cost': 50}, {'id': 46, 'elements': [3, 4], 'cost': 26}, {'id': 47, 'elements': [14, 16, 17], 'cost': 51}, {'id': 48, 'elements': [3, 5, 6, 7], 'cost': 88}, {'id': 49, 'elements': [13, 14, 17], 'cost': 183}, {'id': 50, 'elements': [5, 6, 7, 10, 11], 'cost': 245}, {'id': 51, 'elements': [14, 15, 17], 'cost': 54}, {'id': 52, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 66, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 67, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 68, 'elements': [17], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0003_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0003_bag_stats.png'}","[18, 23, 26, 28, 31, 32, 46]",499.0,"{'num_elements': 17, 'num_sets': 68, 'sets': [{'id': 'S1', 'elements': [13, 15], 'cost': 28}, {'id': 'S2', 'elements': [4, 8, 10], 'cost': 255}, {'id': 'S3', 'elements': [11, 14, 16], 'cost': 156}, {'id': 'S4', 'elements': [1, 3, 6], 'cost': 33}, {'id': 'S5', 'elements': [2, 3, 5], 'cost': 213}, {'id': 'S6', 'elements': [11, 12, 13, 16], 'cost': 52}, {'id': 'S7', 'elements': [2, 3, 4, 5], 'cost': 112}, {'id': 'S8', 'elements': [6, 7, 9], 'cost': 282}, {'id': 'S9', 'elements': [12, 13], 'cost': 40}, {'id': 'S10', 'elements': [12, 13, 14, 16], 'cost': 380}, {'id': 'S11', 'elements': [3, 4, 5], 'cost': 129}, {'id': 'S12', 'elements': [12, 13, 14, 15], 'cost': 372}, {'id': 'S13', 'elements': [8, 9], 'cost': 148}, {'id': 'S14', 'elements': [9, 10, 11, 12, 13], 'cost': 495}, {'id': 'S15', 'elements': [0, 1, 2, 3], 'cost': 140}, {'id': 'S16', 'elements': [7, 8, 9, 11], 'cost': 340}, {'id': 'S17', 'elements': [2, 5, 7], 'cost': 204}, {'id': 'S18', 'elements': [11, 13], 'cost': 80}, {'id': 'S19', 'elements': [0, 1, 3, 4], 'cost': 44}, {'id': 'S20', 'elements': [11, 12, 13], 'cost': 291}, {'id': 'S21', 'elements': [10, 11, 13], 'cost': 189}, {'id': 'S22', 'elements': [6, 9, 10], 'cost': 3}, {'id': 'S23', 'elements': [6, 7], 'cost': 76}, {'id': 'S24', 'elements': [0, 1, 2], 'cost': 207}, {'id': 'S25', 'elements': [13, 14, 15, 16], 'cost': 380}, {'id': 'S26', 'elements': [0, 1], 'cost': 82}, {'id': 'S27', 'elements': [8, 10, 11, 13], 'cost': 168}, {'id': 'S28', 'elements': [15, 16], 'cost': 24}, {'id': 'S29', 'elements': [2, 3, 4], 'cost': 153}, {'id': 'S30', 'elements': [7, 9, 10, 11, 12], 'cost': 125}, {'id': 'S31', 'elements': [8, 9, 10, 12, 14], 'cost': 165}, {'id': 'S32', 'elements': [4, 5], 'cost': 46}, {'id': 'S33', 'elements': [12, 13, 14, 15, 16], 'cost': 415}, {'id': 'S34', 'elements': [7, 8, 10, 12], 'cost': 228}, {'id': 'S35', 'elements': [5, 7, 8, 9], 'cost': 20}, {'id': 'S36', 'elements': [7, 11], 'cost': 24}, {'id': 'S37', 'elements': [5, 6], 'cost': 40}, {'id': 'S38', 'elements': [8, 11], 'cost': 136}, {'id': 'S39', 'elements': [4, 5, 8], 'cost': 300}, {'id': 'S40', 'elements': [0, 2, 3], 'cost': 39}, {'id': 'S41', 'elements': [11, 12, 13, 15], 'cost': 80}, {'id': 'S42', 'elements': [3, 5, 6], 'cost': 255}, {'id': 'S43', 'elements': [1, 2, 4], 'cost': 51}, {'id': 'S44', 'elements': [11, 13, 15, 16], 'cost': 160}, {'id': 'S45', 'elements': [5, 7], 'cost': 50}, {'id': 'S46', 'elements': [2, 3], 'cost': 26}, {'id': 'S47', 'elements': [13, 15, 16], 'cost': 51}, {'id': 'S48', 'elements': [2, 4, 5, 6], 'cost': 88}, {'id': 'S49', 'elements': [12, 13, 16], 'cost': 183}, {'id': 'S50', 'elements': [4, 5, 6, 9, 10], 'cost': 245}, {'id': 'S51', 'elements': [13, 14, 16], 'cost': 54}, {'id': 'S52', 'elements': [0], 'cost': 10000}, {'id': 'S53', 'elements': [1], 'cost': 10000}, {'id': 'S54', 'elements': [2], 'cost': 10000}, {'id': 'S55', 'elements': [3], 'cost': 10000}, {'id': 'S56', 'elements': [4], 'cost': 10000}, {'id': 'S57', 'elements': [5], 'cost': 10000}, {'id': 'S58', 'elements': [6], 'cost': 10000}, {'id': 'S59', 'elements': [7], 'cost': 10000}, {'id': 'S60', 'elements': [8], 'cost': 10000}, {'id': 'S61', 'elements': [9], 'cost': 10000}, {'id': 'S62', 'elements': [10], 'cost': 10000}, {'id': 'S63', 'elements': [11], 'cost': 10000}, {'id': 'S64', 'elements': [12], 'cost': 10000}, {'id': 'S65', 'elements': [13], 'cost': 10000}, {'id': 'S66', 'elements': [14], 'cost': 10000}, {'id': 'S67', 'elements': [15], 'cost': 10000}, {'id': 'S68', 'elements': [16], 'cost': 10000}]}","['S18', 'S23', 'S26', 'S28', 'S31', 'S32', 'S46']",4,json,0
SPP,SPP,"Recently the shipping team was handed a set of pre-made parcel groupings; the task now is to pick which groupings to send so that every package is assigned to a single chosen grouping and nothing is missing or duplicated. Each grouping carries a delivery cost, and the final bill is calculated by adding up the costs of the groupings actually dispatched — the objective is to keep that final number as small as possible. The full list of groupings, which parcels they cover, and their costs is shown below.

{
  ""num_packages"": 15,
  ""num_batches"": 57,
  ""sets"": [
    {
      ""batch_id"": ""S1"",
      ""package_ids"": [
        ""D"",
        ""F"",
        ""H""
      ],
      ""delivery_cost"": 48
    },
    {
      ""batch_id"": ""S2"",
      ""package_ids"": [
        ""K"",
        ""M""
      ],
      ""delivery_cost"": 132
    },
    {
      ""batch_id"": ""S3"",
      ""package_ids"": [
        ""E"",
        ""I""
      ],
      ""delivery_cost"": 116
    },
    {
      ""batch_id"": ""S4"",
      ""package_ids"": [
        ""L"",
        ""M"",
        ""N"",
        ""O""
      ],
      ""delivery_cost"": 44
    },
    {
      ""batch_id"": ""S5"",
      ""package_ids"": [
        ""A"",
        ""B"",
        ""D""
      ],
      ""delivery_cost"": 129
    },
    {
      ""batch_id"": ""S6"",
      ""package_ids"": [
        ""A"",
        ""C"",
        ""D"",
        ""E""
      ],
      ""delivery_cost"": 392
    },
    {
      ""batch_id"": ""S7"",
      ""package_ids"": [
        ""E"",
        ""G"",
        ""H"",
        ""J""
      ],
      ""delivery_cost"": 80
    },
    {
      ""batch_id"": ""S8"",
      ""package_ids"": [
        ""K"",
        ""L"",
        ""N""
      ],
      ""delivery_cost"": 93
    },
    {
      ""batch_id"": ""S9"",
      ""package_ids"": [
        ""A"",
        ""C""
      ],
      ""delivery_cost"": 160
    },
    {
      ""batch_id"": ""S10"",
      ""package_ids"": [
        ""B"",
        ""C"",
        ""D""
      ],
      ""delivery_cost"": 243
    },
    {
      ""batch_id"": ""S11"",
      ""package_ids"": [
        ""A"",
        ""C"",
        ""E""
      ],
      ""delivery_cost"": 138
    },
    {
      ""batch_id"": ""S12"",
      ""package_ids"": [
        ""F"",
        ""H""
      ],
      ""delivery_cost"": 136
    },
    {
      ""batch_id"": ""S13"",
      ""package_ids"": [
        ""B"",
        ""C"",
        ""D"",
        ""E""
      ],
      ""delivery_cost"": 12
    },
    {
      ""batch_id"": ""S14"",
      ""package_ids"": [
        ""E"",
        ""G"",
        ""H""
      ],
      ""delivery_cost"": 288
    },
    {
      ""batch_id"": ""S15"",
      ""package_ids"": [
        ""K"",
        ""L""
      ],
      ""delivery_cost"": 170
    },
    {
      ""batch_id"": ""S16"",
      ""package_ids"": [
        ""K"",
        ""M"",
        ""N""
      ],
      ""delivery_cost"": 138
    },
    {
      ""batch_id"": ""S17"",
      ""package_ids"": [
        ""M"",
        ""N"",
        ""O""
      ],
      ""delivery_cost"": 270
    },
    {
      ""batch_id"": ""S18"",
      ""package_ids"": [
        ""A"",
        ""D"",
        ""E""
      ],
      ""delivery_cost"": 177
    },
    {
      ""batch_id"": ""S19"",
      ""package_ids"": [
        ""B"",
        ""C""
      ],
      ""delivery_cost"": 12
    },
    {
      ""batch_id"": ""S20"",
      ""package_ids"": [
        ""B"",
        ""D"",
        ""F""
      ],
      ""delivery_cost"": 300
    },
    {
      ""batch_id"": ""S21"",
      ""package_ids"": [
        ""I"",
        ""J""
      ],
      ""delivery_cost"": 96
    },
    {
      ""batch_id"": ""S22"",
      ""package_ids"": [
        ""F"",
        ""H"",
        ""I""
      ],
      ""delivery_cost"": 72
    },
    {
      ""batch_id"": ""S23"",
      ""package_ids"": [
        ""G"",
        ""H"",
        ""J""
      ],
      ""delivery_cost"": 33
    },
    {
      ""batch_id"": ""S24"",
      ""package_ids"": [
        ""A"",
        ""B"",
        ""C"",
        ""F""
      ],
      ""delivery_cost"": 56
    },
    {
      ""batch_id"": ""S25"",
      ""package_ids"": [
        ""J"",
        ""K""
      ],
      ""delivery_cost"": 184
    },
    {
      ""batch_id"": ""S26"",
      ""package_ids"": [
        ""I"",
        ""L"",
        ""M"",
        ""N""
      ],
      ""delivery_cost"": 40
    },
    {
      ""batch_id"": ""S27"",
      ""package_ids"": [
        ""F"",
        ""G"",
        ""H"",
        ""J""
      ],
      ""delivery_cost"": 76
    },
    {
      ""batch_id"": ""S28"",
      ""package_ids"": [
        ""H"",
        ""I"",
        ""L"",
        ""M""
      ],
      ""delivery_cost"": 188
    },
    {
      ""batch_id"": ""S29"",
      ""package_ids"": [
        ""G"",
        ""I"",
        ""J""
      ],
      ""delivery_cost"": 144
    },
    {
      ""batch_id"": ""S30"",
      ""package_ids"": [
        ""K"",
        ""L"",
        ""M""
      ],
      ""delivery_cost"": 123
    },
    {
      ""batch_id"": ""S31"",
      ""package_ids"": [
        ""A"",
        ""C"",
        ""G""
      ],
      ""delivery_cost"": 201
    },
    {
      ""batch_id"": ""S32"",
      ""package_ids"": [
        ""I"",
        ""L""
      ],
      ""delivery_cost"": 126
    },
    {
      ""batch_id"": ""S33"",
      ""package_ids"": [
        ""C"",
        ""E""
      ],
      ""delivery_cost"": 184
    },
    {
      ""batch_id"": ""S34"",
      ""package_ids"": [
        ""C"",
        ""E"",
        ""F""
      ],
      ""delivery_cost"": 258
    },
    {
      ""batch_id"": ""S35"",
      ""package_ids"": [
        ""A"",
        ""B"",
        ""C""
      ],
      ""delivery_cost"": 63
    },
    {
      ""batch_id"": ""S36"",
      ""package_ids"": [
        ""J"",
        ""K"",
        ""M"",
        ""N""
      ],
      ""delivery_cost"": 192
    },
    {
      ""batch_id"": ""S37"",
      ""package_ids"": [
        ""G"",
        ""H"",
        ""K""
      ],
      ""delivery_cost"": 153
    },
    {
      ""batch_id"": ""S38"",
      ""package_ids"": [
        ""K"",
        ""L"",
        ""M"",
        ""N""
      ],
      ""delivery_cost"": 68
    },
    {
      ""batch_id"": ""S39"",
      ""package_ids"": [
        ""I"",
        ""J"",
        ""K"",
        ""L""
      ],
      ""delivery_cost"": 400
    },
    {
      ""batch_id"": ""S40"",
      ""package_ids"": [
        ""I"",
        ""J"",
        ""M""
      ],
      ""delivery_cost"": 24
    },
    {
      ""batch_id"": ""S41"",
      ""package_ids"": [
        ""G"",
        ""H"",
        ""I"",
        ""K""
      ],
      ""delivery_cost"": 180
    },
    {
      ""batch_id"": ""S42"",
      ""package_ids"": [
        ""H"",
        ""I"",
        ""J"",
        ""K""
      ],
      ""delivery_cost"": 284
    },
    {
      ""batch_id"": ""S43"",
      ""package_ids"": [
        ""A""
      ],
      ""delivery_cost"": 10000
    },
    {
      ""batch_id"": ""S44"",
      ""package_ids"": [
        ""B""
      ],
      ""delivery_cost"": 10000
    },
    {
      ""batch_id"": ""S45"",
      ""package_ids"": [
        ""C""
      ],
      ""delivery_cost"": 10000
    },
    {
      ""batch_id"": ""S46"",
      ""package_ids"": [
        ""D""
      ],
      ""delivery_cost"": 10000
    },
    {
      ""batch_id"": ""S47"",
      ""package_ids"": [
        ""E""
      ],
      ""delivery_cost"": 10000
    },
    {
      ""batch_id"": ""S48"",
      ""package_ids"": [
        ""F""
      ],
      ""delivery_cost"": 10000
    },
    {
      ""batch_id"": ""S49"",
      ""package_ids"": [
        ""G""
      ],
      ""delivery_cost"": 10000
    },
    {
      ""batch_id"": ""S50"",
      ""package_ids"": [
        ""H""
      ],
      ""delivery_cost"": 10000
    },
    {
      ""batch_id"": ""S51"",
      ""package_ids"": [
        ""I""
      ],
      ""delivery_cost"": 10000
    },
    {
      ""batch_id"": ""S52"",
      ""package_ids"": [
        ""J""
      ],
      ""delivery_cost"": 10000
    },
    {
      ""batch_id"": ""S53"",
      ""package_ids"": [
        ""K""
      ],
      ""delivery_cost"": 10000
    },
    {
      ""batch_id"": ""S54"",
      ""package_ids"": [
        ""L""
      ],
      ""delivery_cost"": 10000
    },
    {
      ""batch_id"": ""S55"",
      ""package_ids"": [
        ""M""
      ],
      ""delivery_cost"": 10000
    },
    {
      ""batch_id"": ""S56"",
      ""package_ids"": [
        ""N""
      ],
      ""delivery_cost"": 10000
    },
    {
      ""batch_id"": ""S57"",
      ""package_ids"": [
        ""O""
      ],
      ""delivery_cost"": 10000
    }
  ]
}

Also, when you send back which groupings we should actually dispatch, please use this simple JSON layout so it's easy to parse:

{
  ""solution"": [""grouping_id"", ...]
}

""solution"" is just a list of the chosen parcel groupings (use the exact identifiers from the instance). This block is just a sketch of the expected shape — not the final answer itself — and each entry in the array should be the identifier of a grouping you want to pick.

Please make sure all identifiers are used exactly as they appear in the instance input — no renaming and no new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'problem_type': 'SPP', 'num_elements': 15, 'num_sets': 57, 'density': 0.16842105263157894, 'sets': [{'id': 1, 'elements': [4, 6, 8], 'cost': 48}, {'id': 2, 'elements': [11, 13], 'cost': 132}, {'id': 3, 'elements': [5, 9], 'cost': 116}, {'id': 4, 'elements': [12, 13, 14, 15], 'cost': 44}, {'id': 5, 'elements': [1, 2, 4], 'cost': 129}, {'id': 6, 'elements': [1, 3, 4, 5], 'cost': 392}, {'id': 7, 'elements': [5, 7, 8, 10], 'cost': 80}, {'id': 8, 'elements': [11, 12, 14], 'cost': 93}, {'id': 9, 'elements': [1, 3], 'cost': 160}, {'id': 10, 'elements': [2, 3, 4], 'cost': 243}, {'id': 11, 'elements': [1, 3, 5], 'cost': 138}, {'id': 12, 'elements': [6, 8], 'cost': 136}, {'id': 13, 'elements': [2, 3, 4, 5], 'cost': 12}, {'id': 14, 'elements': [5, 7, 8], 'cost': 288}, {'id': 15, 'elements': [11, 12], 'cost': 170}, {'id': 16, 'elements': [11, 13, 14], 'cost': 138}, {'id': 17, 'elements': [13, 14, 15], 'cost': 270}, {'id': 18, 'elements': [1, 4, 5], 'cost': 177}, {'id': 19, 'elements': [2, 3], 'cost': 12}, {'id': 20, 'elements': [2, 4, 6], 'cost': 300}, {'id': 21, 'elements': [9, 10], 'cost': 96}, {'id': 22, 'elements': [6, 8, 9], 'cost': 72}, {'id': 23, 'elements': [7, 8, 10], 'cost': 33}, {'id': 24, 'elements': [1, 2, 3, 6], 'cost': 56}, {'id': 25, 'elements': [10, 11], 'cost': 184}, {'id': 26, 'elements': [9, 12, 13, 14], 'cost': 40}, {'id': 27, 'elements': [6, 7, 8, 10], 'cost': 76}, {'id': 28, 'elements': [8, 9, 12, 13], 'cost': 188}, {'id': 29, 'elements': [7, 9, 10], 'cost': 144}, {'id': 30, 'elements': [11, 12, 13], 'cost': 123}, {'id': 31, 'elements': [1, 3, 7], 'cost': 201}, {'id': 32, 'elements': [9, 12], 'cost': 126}, {'id': 33, 'elements': [3, 5], 'cost': 184}, {'id': 34, 'elements': [3, 5, 6], 'cost': 258}, {'id': 35, 'elements': [1, 2, 3], 'cost': 63}, {'id': 36, 'elements': [10, 11, 13, 14], 'cost': 192}, {'id': 37, 'elements': [7, 8, 11], 'cost': 153}, {'id': 38, 'elements': [11, 12, 13, 14], 'cost': 68}, {'id': 39, 'elements': [9, 10, 11, 12], 'cost': 400}, {'id': 40, 'elements': [9, 10, 13], 'cost': 24}, {'id': 41, 'elements': [7, 8, 9, 11], 'cost': 180}, {'id': 42, 'elements': [8, 9, 10, 11], 'cost': 284}, {'id': 43, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 44, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 45, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 46, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 47, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [15], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0004_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0004_bag_stats.png'}","[4, 5, 21, 34, 37]",680.0,"{'num_elements': 15, 'num_sets': 57, 'sets': [{'id': 'S1', 'elements': ['D', 'F', 'H'], 'cost': 48}, {'id': 'S2', 'elements': ['K', 'M'], 'cost': 132}, {'id': 'S3', 'elements': ['E', 'I'], 'cost': 116}, {'id': 'S4', 'elements': ['L', 'M', 'N', 'O'], 'cost': 44}, {'id': 'S5', 'elements': ['A', 'B', 'D'], 'cost': 129}, {'id': 'S6', 'elements': ['A', 'C', 'D', 'E'], 'cost': 392}, {'id': 'S7', 'elements': ['E', 'G', 'H', 'J'], 'cost': 80}, {'id': 'S8', 'elements': ['K', 'L', 'N'], 'cost': 93}, {'id': 'S9', 'elements': ['A', 'C'], 'cost': 160}, {'id': 'S10', 'elements': ['B', 'C', 'D'], 'cost': 243}, {'id': 'S11', 'elements': ['A', 'C', 'E'], 'cost': 138}, {'id': 'S12', 'elements': ['F', 'H'], 'cost': 136}, {'id': 'S13', 'elements': ['B', 'C', 'D', 'E'], 'cost': 12}, {'id': 'S14', 'elements': ['E', 'G', 'H'], 'cost': 288}, {'id': 'S15', 'elements': ['K', 'L'], 'cost': 170}, {'id': 'S16', 'elements': ['K', 'M', 'N'], 'cost': 138}, {'id': 'S17', 'elements': ['M', 'N', 'O'], 'cost': 270}, {'id': 'S18', 'elements': ['A', 'D', 'E'], 'cost': 177}, {'id': 'S19', 'elements': ['B', 'C'], 'cost': 12}, {'id': 'S20', 'elements': ['B', 'D', 'F'], 'cost': 300}, {'id': 'S21', 'elements': ['I', 'J'], 'cost': 96}, {'id': 'S22', 'elements': ['F', 'H', 'I'], 'cost': 72}, {'id': 'S23', 'elements': ['G', 'H', 'J'], 'cost': 33}, {'id': 'S24', 'elements': ['A', 'B', 'C', 'F'], 'cost': 56}, {'id': 'S25', 'elements': ['J', 'K'], 'cost': 184}, {'id': 'S26', 'elements': ['I', 'L', 'M', 'N'], 'cost': 40}, {'id': 'S27', 'elements': ['F', 'G', 'H', 'J'], 'cost': 76}, {'id': 'S28', 'elements': ['H', 'I', 'L', 'M'], 'cost': 188}, {'id': 'S29', 'elements': ['G', 'I', 'J'], 'cost': 144}, {'id': 'S30', 'elements': ['K', 'L', 'M'], 'cost': 123}, {'id': 'S31', 'elements': ['A', 'C', 'G'], 'cost': 201}, {'id': 'S32', 'elements': ['I', 'L'], 'cost': 126}, {'id': 'S33', 'elements': ['C', 'E'], 'cost': 184}, {'id': 'S34', 'elements': ['C', 'E', 'F'], 'cost': 258}, {'id': 'S35', 'elements': ['A', 'B', 'C'], 'cost': 63}, {'id': 'S36', 'elements': ['J', 'K', 'M', 'N'], 'cost': 192}, {'id': 'S37', 'elements': ['G', 'H', 'K'], 'cost': 153}, {'id': 'S38', 'elements': ['K', 'L', 'M', 'N'], 'cost': 68}, {'id': 'S39', 'elements': ['I', 'J', 'K', 'L'], 'cost': 400}, {'id': 'S40', 'elements': ['I', 'J', 'M'], 'cost': 24}, {'id': 'S41', 'elements': ['G', 'H', 'I', 'K'], 'cost': 180}, {'id': 'S42', 'elements': ['H', 'I', 'J', 'K'], 'cost': 284}, {'id': 'S43', 'elements': ['A'], 'cost': 10000}, {'id': 'S44', 'elements': ['B'], 'cost': 10000}, {'id': 'S45', 'elements': ['C'], 'cost': 10000}, {'id': 'S46', 'elements': ['D'], 'cost': 10000}, {'id': 'S47', 'elements': ['E'], 'cost': 10000}, {'id': 'S48', 'elements': ['F'], 'cost': 10000}, {'id': 'S49', 'elements': ['G'], 'cost': 10000}, {'id': 'S50', 'elements': ['H'], 'cost': 10000}, {'id': 'S51', 'elements': ['I'], 'cost': 10000}, {'id': 'S52', 'elements': ['J'], 'cost': 10000}, {'id': 'S53', 'elements': ['K'], 'cost': 10000}, {'id': 'S54', 'elements': ['L'], 'cost': 10000}, {'id': 'S55', 'elements': ['M'], 'cost': 10000}, {'id': 'S56', 'elements': ['N'], 'cost': 10000}, {'id': 'S57', 'elements': ['O'], 'cost': 10000}]}","['S4', 'S5', 'S21', 'S34', 'S37']",5,json,names
SPP,SPP,"We’ve got a patch of land with a bunch of species to place and a handful of ready-made beds to choose from, each with its own cost to prepare. The choice to make is which beds to prepare so every species is assigned to one and only one of the beds we pick — no species missing and no doubling up. The way to judge different choices is by adding up the setup fees for the beds we end up using and preferring the plan with the smallest total. The exact bed options, what they cover, and their prices are listed below.

- **num_species**: 17
- **num_beds**: 83

| bed_id | bed_setup_cost | species_in_bed |
|---|---|---|
| S1 | 292 | 2 3 5 7 |
| S2 | 198 | 0 3 |
| S3 | 324 | 4 5 6 7 |
| S4 | 40 | 7 9 10 11 |
| S5 | 164 | 3 4 5 7 |
| S6 | 360 | 8 9 11 12 14 |
| S7 | 64 | 6 7 |
| S8 | 158 | 1 2 |
| S9 | 75 | 5 6 8 |
| S10 | 48 | 13 14 15 16 |
| S11 | 385 | 11 13 14 15 16 |
| S12 | 75 | 11 12 13 |
| S13 | 108 | 7 9 11 |
| S14 | 235 | 1 2 3 4 7 |
| S15 | 368 | 3 4 5 6 |
| S16 | 54 | 13 15 16 |
| S17 | 117 | 8 11 14 |
| S18 | 372 | 9 10 11 12 |
| S19 | 150 | 0 1 2 3 4 |
| S20 | 30 | 4 10 |
| S21 | 178 | 12 13 |
| S22 | 190 | 15 16 |
| S23 | 196 | 3 4 5 9 |
| S24 | 261 | 4 5 6 |
| S25 | 116 | 6 7 8 9 |
| S26 | 360 | 7 8 9 10 11 |
| S27 | 225 | 6 7 8 9 11 |
| S28 | 165 | 2 3 4 5 6 |
| S29 | 72 | 5 6 7 9 |
| S30 | 136 | 6 8 |
| S31 | 184 | 4 7 |
| S32 | 159 | 0 4 7 |
| S33 | 20 | 1 3 4 5 |
| S34 | 320 | 2 5 6 7 8 |
| S35 | 146 | 5 11 |
| S36 | 72 | 9 10 11 14 |
| S37 | 460 | 12 13 14 15 16 |
| S38 | 194 | 10 11 |
| S39 | 24 | 12 14 |
| S40 | 96 | 1 2 3 |
| S41 | 86 | 11 13 |
| S42 | 258 | 0 1 2 |
| S43 | 182 | 7 10 |
| S44 | 72 | 1 2 4 6 |
| S45 | 213 | 13 14 15 |
| S46 | 340 | 4 5 6 7 8 |
| S47 | 104 | 0 3 4 5 |
| S48 | 368 | 2 4 5 6 |
| S49 | 216 | 0 1 2 4 |
| S50 | 150 | 9 10 |
| S51 | 16 | 10 12 |
| S52 | 415 | 0 1 2 3 6 |
| S53 | 55 | 11 12 13 15 16 |
| S54 | 272 | 2 3 4 5 |
| S55 | 78 | 8 9 |
| S56 | 135 | 11 13 15 |
| S57 | 30 | 7 9 |
| S58 | 6 | 13 16 |
| S59 | 273 | 0 1 3 |
| S60 | 40 | 5 6 7 8 10 |
| S61 | 148 | 8 12 |
| S62 | 201 | 9 12 14 |
| S63 | 152 | 8 10 |
| S64 | 430 | 4 5 6 8 9 |
| S65 | 57 | 13 14 16 |
| S66 | 144 | 3 5 6 |
| S67 | 10000 | 0 |
| S68 | 10000 | 1 |
| S69 | 10000 | 2 |
| S70 | 10000 | 3 |
| S71 | 10000 | 4 |
| S72 | 10000 | 5 |
| S73 | 10000 | 6 |
| S74 | 10000 | 7 |
| S75 | 10000 | 8 |
| S76 | 10000 | 9 |
| S77 | 10000 | 10 |
| S78 | 10000 | 11 |
| S79 | 10000 | 12 |
| S80 | 10000 | 13 |
| S81 | 10000 | 14 |
| S82 | 10000 | 15 |
| S83 | 10000 | 16 |

If you want to hand me the chosen beds in a simple, machine-friendly way, just return them in a tiny JSON snippet like this:

{
  ""solution"": [""bed_id"", ...]
}

""solution"" is the list of beds to prepare; each ""bed_id"" is a placeholder for one of the bed identifiers from the instance (replace it with the actual bed ID). This is just the expected shape — a sketch of how I want the answer formatted, not the plan itself.

Please use the exact identifiers from the instance input — don't rename them or invent new labels. 
Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'problem_type': 'SPP', 'num_elements': 17, 'num_sets': 83, 'density': 0.1700921332388377, 'sets': [{'id': 1, 'elements': [3, 4, 6, 8], 'cost': 292}, {'id': 2, 'elements': [1, 4], 'cost': 198}, {'id': 3, 'elements': [5, 6, 7, 8], 'cost': 324}, {'id': 4, 'elements': [8, 10, 11, 12], 'cost': 40}, {'id': 5, 'elements': [4, 5, 6, 8], 'cost': 164}, {'id': 6, 'elements': [9, 10, 12, 13, 15], 'cost': 360}, {'id': 7, 'elements': [7, 8], 'cost': 64}, {'id': 8, 'elements': [2, 3], 'cost': 158}, {'id': 9, 'elements': [6, 7, 9], 'cost': 75}, {'id': 10, 'elements': [14, 15, 16, 17], 'cost': 48}, {'id': 11, 'elements': [12, 14, 15, 16, 17], 'cost': 385}, {'id': 12, 'elements': [12, 13, 14], 'cost': 75}, {'id': 13, 'elements': [8, 10, 12], 'cost': 108}, {'id': 14, 'elements': [2, 3, 4, 5, 8], 'cost': 235}, {'id': 15, 'elements': [4, 5, 6, 7], 'cost': 368}, {'id': 16, 'elements': [14, 16, 17], 'cost': 54}, {'id': 17, 'elements': [9, 12, 15], 'cost': 117}, {'id': 18, 'elements': [10, 11, 12, 13], 'cost': 372}, {'id': 19, 'elements': [1, 2, 3, 4, 5], 'cost': 150}, {'id': 20, 'elements': [5, 11], 'cost': 30}, {'id': 21, 'elements': [13, 14], 'cost': 178}, {'id': 22, 'elements': [16, 17], 'cost': 190}, {'id': 23, 'elements': [4, 5, 6, 10], 'cost': 196}, {'id': 24, 'elements': [5, 6, 7], 'cost': 261}, {'id': 25, 'elements': [7, 8, 9, 10], 'cost': 116}, {'id': 26, 'elements': [8, 9, 10, 11, 12], 'cost': 360}, {'id': 27, 'elements': [7, 8, 9, 10, 12], 'cost': 225}, {'id': 28, 'elements': [3, 4, 5, 6, 7], 'cost': 165}, {'id': 29, 'elements': [6, 7, 8, 10], 'cost': 72}, {'id': 30, 'elements': [7, 9], 'cost': 136}, {'id': 31, 'elements': [5, 8], 'cost': 184}, {'id': 32, 'elements': [1, 5, 8], 'cost': 159}, {'id': 33, 'elements': [2, 4, 5, 6], 'cost': 20}, {'id': 34, 'elements': [3, 6, 7, 8, 9], 'cost': 320}, {'id': 35, 'elements': [6, 12], 'cost': 146}, {'id': 36, 'elements': [10, 11, 12, 15], 'cost': 72}, {'id': 37, 'elements': [13, 14, 15, 16, 17], 'cost': 460}, {'id': 38, 'elements': [11, 12], 'cost': 194}, {'id': 39, 'elements': [13, 15], 'cost': 24}, {'id': 40, 'elements': [2, 3, 4], 'cost': 96}, {'id': 41, 'elements': [12, 14], 'cost': 86}, {'id': 42, 'elements': [1, 2, 3], 'cost': 258}, {'id': 43, 'elements': [8, 11], 'cost': 182}, {'id': 44, 'elements': [2, 3, 5, 7], 'cost': 72}, {'id': 45, 'elements': [14, 15, 16], 'cost': 213}, {'id': 46, 'elements': [5, 6, 7, 8, 9], 'cost': 340}, {'id': 47, 'elements': [1, 4, 5, 6], 'cost': 104}, {'id': 48, 'elements': [3, 5, 6, 7], 'cost': 368}, {'id': 49, 'elements': [1, 2, 3, 5], 'cost': 216}, {'id': 50, 'elements': [10, 11], 'cost': 150}, {'id': 51, 'elements': [11, 13], 'cost': 16}, {'id': 52, 'elements': [1, 2, 3, 4, 7], 'cost': 415}, {'id': 53, 'elements': [12, 13, 14, 16, 17], 'cost': 55}, {'id': 54, 'elements': [3, 4, 5, 6], 'cost': 272}, {'id': 55, 'elements': [9, 10], 'cost': 78}, {'id': 56, 'elements': [12, 14, 16], 'cost': 135}, {'id': 57, 'elements': [8, 10], 'cost': 30}, {'id': 58, 'elements': [14, 17], 'cost': 6}, {'id': 59, 'elements': [1, 2, 4], 'cost': 273}, {'id': 60, 'elements': [6, 7, 8, 9, 11], 'cost': 40}, {'id': 61, 'elements': [9, 13], 'cost': 148}, {'id': 62, 'elements': [10, 13, 15], 'cost': 201}, {'id': 63, 'elements': [9, 11], 'cost': 152}, {'id': 64, 'elements': [5, 6, 7, 9, 10], 'cost': 430}, {'id': 65, 'elements': [14, 15, 17], 'cost': 57}, {'id': 66, 'elements': [4, 6, 7], 'cost': 144}, {'id': 67, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 68, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 69, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 70, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 71, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 72, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 73, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 74, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 75, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 76, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 77, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 78, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 79, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 80, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 81, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 82, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 83, 'elements': [17], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0005_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0005_bag_stats.png'}","[4, 9, 16, 19, 39]",343.0,"{'num_elements': 17, 'num_sets': 83, 'sets': [{'id': 'S1', 'elements': [2, 3, 5, 7], 'cost': 292}, {'id': 'S2', 'elements': [0, 3], 'cost': 198}, {'id': 'S3', 'elements': [4, 5, 6, 7], 'cost': 324}, {'id': 'S4', 'elements': [7, 9, 10, 11], 'cost': 40}, {'id': 'S5', 'elements': [3, 4, 5, 7], 'cost': 164}, {'id': 'S6', 'elements': [8, 9, 11, 12, 14], 'cost': 360}, {'id': 'S7', 'elements': [6, 7], 'cost': 64}, {'id': 'S8', 'elements': [1, 2], 'cost': 158}, {'id': 'S9', 'elements': [5, 6, 8], 'cost': 75}, {'id': 'S10', 'elements': [13, 14, 15, 16], 'cost': 48}, {'id': 'S11', 'elements': [11, 13, 14, 15, 16], 'cost': 385}, {'id': 'S12', 'elements': [11, 12, 13], 'cost': 75}, {'id': 'S13', 'elements': [7, 9, 11], 'cost': 108}, {'id': 'S14', 'elements': [1, 2, 3, 4, 7], 'cost': 235}, {'id': 'S15', 'elements': [3, 4, 5, 6], 'cost': 368}, {'id': 'S16', 'elements': [13, 15, 16], 'cost': 54}, {'id': 'S17', 'elements': [8, 11, 14], 'cost': 117}, {'id': 'S18', 'elements': [9, 10, 11, 12], 'cost': 372}, {'id': 'S19', 'elements': [0, 1, 2, 3, 4], 'cost': 150}, {'id': 'S20', 'elements': [4, 10], 'cost': 30}, {'id': 'S21', 'elements': [12, 13], 'cost': 178}, {'id': 'S22', 'elements': [15, 16], 'cost': 190}, {'id': 'S23', 'elements': [3, 4, 5, 9], 'cost': 196}, {'id': 'S24', 'elements': [4, 5, 6], 'cost': 261}, {'id': 'S25', 'elements': [6, 7, 8, 9], 'cost': 116}, {'id': 'S26', 'elements': [7, 8, 9, 10, 11], 'cost': 360}, {'id': 'S27', 'elements': [6, 7, 8, 9, 11], 'cost': 225}, {'id': 'S28', 'elements': [2, 3, 4, 5, 6], 'cost': 165}, {'id': 'S29', 'elements': [5, 6, 7, 9], 'cost': 72}, {'id': 'S30', 'elements': [6, 8], 'cost': 136}, {'id': 'S31', 'elements': [4, 7], 'cost': 184}, {'id': 'S32', 'elements': [0, 4, 7], 'cost': 159}, {'id': 'S33', 'elements': [1, 3, 4, 5], 'cost': 20}, {'id': 'S34', 'elements': [2, 5, 6, 7, 8], 'cost': 320}, {'id': 'S35', 'elements': [5, 11], 'cost': 146}, {'id': 'S36', 'elements': [9, 10, 11, 14], 'cost': 72}, {'id': 'S37', 'elements': [12, 13, 14, 15, 16], 'cost': 460}, {'id': 'S38', 'elements': [10, 11], 'cost': 194}, {'id': 'S39', 'elements': [12, 14], 'cost': 24}, {'id': 'S40', 'elements': [1, 2, 3], 'cost': 96}, {'id': 'S41', 'elements': [11, 13], 'cost': 86}, {'id': 'S42', 'elements': [0, 1, 2], 'cost': 258}, {'id': 'S43', 'elements': [7, 10], 'cost': 182}, {'id': 'S44', 'elements': [1, 2, 4, 6], 'cost': 72}, {'id': 'S45', 'elements': [13, 14, 15], 'cost': 213}, {'id': 'S46', 'elements': [4, 5, 6, 7, 8], 'cost': 340}, {'id': 'S47', 'elements': [0, 3, 4, 5], 'cost': 104}, {'id': 'S48', 'elements': [2, 4, 5, 6], 'cost': 368}, {'id': 'S49', 'elements': [0, 1, 2, 4], 'cost': 216}, {'id': 'S50', 'elements': [9, 10], 'cost': 150}, {'id': 'S51', 'elements': [10, 12], 'cost': 16}, {'id': 'S52', 'elements': [0, 1, 2, 3, 6], 'cost': 415}, {'id': 'S53', 'elements': [11, 12, 13, 15, 16], 'cost': 55}, {'id': 'S54', 'elements': [2, 3, 4, 5], 'cost': 272}, {'id': 'S55', 'elements': [8, 9], 'cost': 78}, {'id': 'S56', 'elements': [11, 13, 15], 'cost': 135}, {'id': 'S57', 'elements': [7, 9], 'cost': 30}, {'id': 'S58', 'elements': [13, 16], 'cost': 6}, {'id': 'S59', 'elements': [0, 1, 3], 'cost': 273}, {'id': 'S60', 'elements': [5, 6, 7, 8, 10], 'cost': 40}, {'id': 'S61', 'elements': [8, 12], 'cost': 148}, {'id': 'S62', 'elements': [9, 12, 14], 'cost': 201}, {'id': 'S63', 'elements': [8, 10], 'cost': 152}, {'id': 'S64', 'elements': [4, 5, 6, 8, 9], 'cost': 430}, {'id': 'S65', 'elements': [13, 14, 16], 'cost': 57}, {'id': 'S66', 'elements': [3, 5, 6], 'cost': 144}, {'id': 'S67', 'elements': [0], 'cost': 10000}, {'id': 'S68', 'elements': [1], 'cost': 10000}, {'id': 'S69', 'elements': [2], 'cost': 10000}, {'id': 'S70', 'elements': [3], 'cost': 10000}, {'id': 'S71', 'elements': [4], 'cost': 10000}, {'id': 'S72', 'elements': [5], 'cost': 10000}, {'id': 'S73', 'elements': [6], 'cost': 10000}, {'id': 'S74', 'elements': [7], 'cost': 10000}, {'id': 'S75', 'elements': [8], 'cost': 10000}, {'id': 'S76', 'elements': [9], 'cost': 10000}, {'id': 'S77', 'elements': [10], 'cost': 10000}, {'id': 'S78', 'elements': [11], 'cost': 10000}, {'id': 'S79', 'elements': [12], 'cost': 10000}, {'id': 'S80', 'elements': [13], 'cost': 10000}, {'id': 'S81', 'elements': [14], 'cost': 10000}, {'id': 'S82', 'elements': [15], 'cost': 10000}, {'id': 'S83', 'elements': [16], 'cost': 10000}]}","['S4', 'S9', 'S16', 'S19', 'S39']",6,markdown_table,0
SPP,SPP,"There’s a seasonal promotion being planned where a bunch of garments need to be grouped into preset outfit bundles. The decision is which bundles to actually offer so that each garment is covered by one, and only one, chosen bundle. The winning plan is simply the one with the smallest combined price — just sum the prices of the bundles that are chosen — and every item must be included exactly once. The full list of items and bundle choices is shown below.

# total_garments=13
# total_bundles_available=45
bundle_id,bundle_price,garments_in_bundle
S1,30,0 1 2
S2,56,11 12
S3,75,4 5 7
S4,364,0 1 2 3
S5,98,3 4
S6,192,9 10 11
S7,81,10 11 12
S8,190,4 7
S9,186,4 5
S10,146,1 2
S11,294,6 7 9
S12,210,1 2 3
S13,81,0
S14,118,10 11
S15,124,9 11
S16,270,3 4 5
S17,105,2 3 5
S18,336,1 2 4 6
S19,24,2 3 4 5
S20,164,7 10 11 12
S21,97,3
S22,219,5 6 8
S23,285,5 6 7
S24,105,6 9 10
S25,198,9 10
S26,114,0 3
S27,112,2 3
S28,24,6 8
S29,128,0 1 2 4
S30,108,0 2
S31,86,1 4
S32,267,9 10 12
S33,10000,0
S34,10000,1
S35,10000,2
S36,10000,3
S37,10000,4
S38,10000,5
S39,10000,6
S40,10000,7
S41,10000,8
S42,10000,9
S43,10000,10
S44,10000,11
S45,10000,12

Also, when you send the final selection, please stick to a tiny JSON layout like this:

{
  ""solution"": [""bundle_id"", ...]
}

Here ""solution"" is the list of the outfit bundle IDs you choose — each entry should be the exact bundle identifier from the instance (the placeholder ""bundle_id"" just shows the kind of label to use). The ellipsis means you can list as many bundle IDs as needed. This is just a sketch of the shape I want, not the actual answer.

Please make sure you use the identifiers exactly as they appear in the instance input — no renaming and no new labels.

For example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 13, 'num_sets': 45, 'density': 0.1658119658119658, 'sets': [{'id': 1, 'elements': [1, 2, 3], 'cost': 30}, {'id': 2, 'elements': [12, 13], 'cost': 56}, {'id': 3, 'elements': [5, 6, 8], 'cost': 75}, {'id': 4, 'elements': [1, 2, 3, 4], 'cost': 364}, {'id': 5, 'elements': [4, 5], 'cost': 98}, {'id': 6, 'elements': [10, 11, 12], 'cost': 192}, {'id': 7, 'elements': [11, 12, 13], 'cost': 81}, {'id': 8, 'elements': [5, 8], 'cost': 190}, {'id': 9, 'elements': [5, 6], 'cost': 186}, {'id': 10, 'elements': [2, 3], 'cost': 146}, {'id': 11, 'elements': [7, 8, 10], 'cost': 294}, {'id': 12, 'elements': [2, 3, 4], 'cost': 210}, {'id': 13, 'elements': [1], 'cost': 81}, {'id': 14, 'elements': [11, 12], 'cost': 118}, {'id': 15, 'elements': [10, 12], 'cost': 124}, {'id': 16, 'elements': [4, 5, 6], 'cost': 270}, {'id': 17, 'elements': [3, 4, 6], 'cost': 105}, {'id': 18, 'elements': [2, 3, 5, 7], 'cost': 336}, {'id': 19, 'elements': [3, 4, 5, 6], 'cost': 24}, {'id': 20, 'elements': [8, 11, 12, 13], 'cost': 164}, {'id': 21, 'elements': [4], 'cost': 97}, {'id': 22, 'elements': [6, 7, 9], 'cost': 219}, {'id': 23, 'elements': [6, 7, 8], 'cost': 285}, {'id': 24, 'elements': [7, 10, 11], 'cost': 105}, {'id': 25, 'elements': [10, 11], 'cost': 198}, {'id': 26, 'elements': [1, 4], 'cost': 114}, {'id': 27, 'elements': [3, 4], 'cost': 112}, {'id': 28, 'elements': [7, 9], 'cost': 24}, {'id': 29, 'elements': [1, 2, 3, 5], 'cost': 128}, {'id': 30, 'elements': [1, 3], 'cost': 108}, {'id': 31, 'elements': [2, 5], 'cost': 86}, {'id': 32, 'elements': [10, 11, 13], 'cost': 267}, {'id': 33, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 34, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 35, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 36, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 37, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 38, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 39, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 40, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 41, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 42, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 43, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 44, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 45, 'elements': [13], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0006_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0006_bag_stats.png'}","[1, 2, 3, 21, 25, 28]",480.0,"{'num_elements': 13, 'num_sets': 45, 'sets': [{'id': 'S1', 'elements': [0, 1, 2], 'cost': 30}, {'id': 'S2', 'elements': [11, 12], 'cost': 56}, {'id': 'S3', 'elements': [4, 5, 7], 'cost': 75}, {'id': 'S4', 'elements': [0, 1, 2, 3], 'cost': 364}, {'id': 'S5', 'elements': [3, 4], 'cost': 98}, {'id': 'S6', 'elements': [9, 10, 11], 'cost': 192}, {'id': 'S7', 'elements': [10, 11, 12], 'cost': 81}, {'id': 'S8', 'elements': [4, 7], 'cost': 190}, {'id': 'S9', 'elements': [4, 5], 'cost': 186}, {'id': 'S10', 'elements': [1, 2], 'cost': 146}, {'id': 'S11', 'elements': [6, 7, 9], 'cost': 294}, {'id': 'S12', 'elements': [1, 2, 3], 'cost': 210}, {'id': 'S13', 'elements': [0], 'cost': 81}, {'id': 'S14', 'elements': [10, 11], 'cost': 118}, {'id': 'S15', 'elements': [9, 11], 'cost': 124}, {'id': 'S16', 'elements': [3, 4, 5], 'cost': 270}, {'id': 'S17', 'elements': [2, 3, 5], 'cost': 105}, {'id': 'S18', 'elements': [1, 2, 4, 6], 'cost': 336}, {'id': 'S19', 'elements': [2, 3, 4, 5], 'cost': 24}, {'id': 'S20', 'elements': [7, 10, 11, 12], 'cost': 164}, {'id': 'S21', 'elements': [3], 'cost': 97}, {'id': 'S22', 'elements': [5, 6, 8], 'cost': 219}, {'id': 'S23', 'elements': [5, 6, 7], 'cost': 285}, {'id': 'S24', 'elements': [6, 9, 10], 'cost': 105}, {'id': 'S25', 'elements': [9, 10], 'cost': 198}, {'id': 'S26', 'elements': [0, 3], 'cost': 114}, {'id': 'S27', 'elements': [2, 3], 'cost': 112}, {'id': 'S28', 'elements': [6, 8], 'cost': 24}, {'id': 'S29', 'elements': [0, 1, 2, 4], 'cost': 128}, {'id': 'S30', 'elements': [0, 2], 'cost': 108}, {'id': 'S31', 'elements': [1, 4], 'cost': 86}, {'id': 'S32', 'elements': [9, 10, 12], 'cost': 267}, {'id': 'S33', 'elements': [0], 'cost': 10000}, {'id': 'S34', 'elements': [1], 'cost': 10000}, {'id': 'S35', 'elements': [2], 'cost': 10000}, {'id': 'S36', 'elements': [3], 'cost': 10000}, {'id': 'S37', 'elements': [4], 'cost': 10000}, {'id': 'S38', 'elements': [5], 'cost': 10000}, {'id': 'S39', 'elements': [6], 'cost': 10000}, {'id': 'S40', 'elements': [7], 'cost': 10000}, {'id': 'S41', 'elements': [8], 'cost': 10000}, {'id': 'S42', 'elements': [9], 'cost': 10000}, {'id': 'S43', 'elements': [10], 'cost': 10000}, {'id': 'S44', 'elements': [11], 'cost': 10000}, {'id': 'S45', 'elements': [12], 'cost': 10000}]}","['S1', 'S2', 'S3', 'S21', 'S25', 'S28']",7,csv,0
SPP,SPP,"Someone at the branch wants the shelves reorganized so each book sits in a single selected shelf group and no title ends up unassigned or assigned twice. The trick is choosing which groups to use so the combined cost is as small as it can be — simply add together the costs of the groups you pick to see the total. The detailed options and prices follow below.

# num_titles=15
# num_shelf_groups_available=55
shelf_group_id,shelf_group_cost,titles_in_group
S1,12,I K N
S2,36,G H I J
S3,132,K L M N
S4,231,K L M
S5,162,G H I
S6,231,M N O
S7,66,A B D
S8,291,A B C
S9,68,I M
S10,148,L M
S11,152,F G
S12,44,A B C D
S13,118,J L
S14,164,H I J K
S15,88,A C
S16,90,H J L
S17,316,H J K L
S18,388,F G H J
S19,168,K L M O
S20,136,B D
S21,236,D E F G
S22,72,I J K
S23,69,C D F
S24,240,F G J
S25,50,F H
S26,20,F G I J
S27,146,H J
S28,171,F G H
S29,117,E H I
S30,364,L M N O
S31,81,E F G
S32,52,D E G H
S33,72,N O
S34,148,M N
S35,297,J K L
S36,96,I J
S37,282,D G I
S38,18,H I
S39,148,E F G H
S40,264,H K L
S41,10000,A
S42,10000,B
S43,10000,C
S44,10000,D
S45,10000,E
S46,10000,F
S47,10000,G
S48,10000,H
S49,10000,I
S50,10000,J
S51,10000,K
S52,10000,L
S53,10000,M
S54,10000,N
S55,10000,O

If you want to tell me which shelf groups to pick, just send it back in this little JSON shape so I can read it cleanly:

{
  ""solution"": [""shelf_group_id"", ...]
}

The ""solution"" array is where you list the shelf group identifiers you choose. Each entry in that list is a placeholder for a single selected shelf group (think of it like the group name or code for the shelves where each book will live). This is just a sketch of the shape I expect, not the actual answer — replace those placeholders with the real IDs from the instance.

Please make sure to use the exact identifiers given in the instance input — do not rename them or invent new labels. 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 15, 'num_sets': 55, 'density': 0.16363636363636364, 'sets': [{'id': 1, 'elements': [9, 11, 14], 'cost': 12}, {'id': 2, 'elements': [7, 8, 9, 10], 'cost': 36}, {'id': 3, 'elements': [11, 12, 13, 14], 'cost': 132}, {'id': 4, 'elements': [11, 12, 13], 'cost': 231}, {'id': 5, 'elements': [7, 8, 9], 'cost': 162}, {'id': 6, 'elements': [13, 14, 15], 'cost': 231}, {'id': 7, 'elements': [1, 2, 4], 'cost': 66}, {'id': 8, 'elements': [1, 2, 3], 'cost': 291}, {'id': 9, 'elements': [9, 13], 'cost': 68}, {'id': 10, 'elements': [12, 13], 'cost': 148}, {'id': 11, 'elements': [6, 7], 'cost': 152}, {'id': 12, 'elements': [1, 2, 3, 4], 'cost': 44}, {'id': 13, 'elements': [10, 12], 'cost': 118}, {'id': 14, 'elements': [8, 9, 10, 11], 'cost': 164}, {'id': 15, 'elements': [1, 3], 'cost': 88}, {'id': 16, 'elements': [8, 10, 12], 'cost': 90}, {'id': 17, 'elements': [8, 10, 11, 12], 'cost': 316}, {'id': 18, 'elements': [6, 7, 8, 10], 'cost': 388}, {'id': 19, 'elements': [11, 12, 13, 15], 'cost': 168}, {'id': 20, 'elements': [2, 4], 'cost': 136}, {'id': 21, 'elements': [4, 5, 6, 7], 'cost': 236}, {'id': 22, 'elements': [9, 10, 11], 'cost': 72}, {'id': 23, 'elements': [3, 4, 6], 'cost': 69}, {'id': 24, 'elements': [6, 7, 10], 'cost': 240}, {'id': 25, 'elements': [6, 8], 'cost': 50}, {'id': 26, 'elements': [6, 7, 9, 10], 'cost': 20}, {'id': 27, 'elements': [8, 10], 'cost': 146}, {'id': 28, 'elements': [6, 7, 8], 'cost': 171}, {'id': 29, 'elements': [5, 8, 9], 'cost': 117}, {'id': 30, 'elements': [12, 13, 14, 15], 'cost': 364}, {'id': 31, 'elements': [5, 6, 7], 'cost': 81}, {'id': 32, 'elements': [4, 5, 7, 8], 'cost': 52}, {'id': 33, 'elements': [14, 15], 'cost': 72}, {'id': 34, 'elements': [13, 14], 'cost': 148}, {'id': 35, 'elements': [10, 11, 12], 'cost': 297}, {'id': 36, 'elements': [9, 10], 'cost': 96}, {'id': 37, 'elements': [4, 7, 9], 'cost': 282}, {'id': 38, 'elements': [8, 9], 'cost': 18}, {'id': 39, 'elements': [5, 6, 7, 8], 'cost': 148}, {'id': 40, 'elements': [8, 11, 12], 'cost': 264}, {'id': 41, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 42, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 43, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 44, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 45, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 46, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 47, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [15], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0007_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0007_bag_stats.png'}","[10, 12, 22, 33, 39]",484.0,"{'num_elements': 15, 'num_sets': 55, 'sets': [{'id': 'S1', 'elements': ['I', 'K', 'N'], 'cost': 12}, {'id': 'S2', 'elements': ['G', 'H', 'I', 'J'], 'cost': 36}, {'id': 'S3', 'elements': ['K', 'L', 'M', 'N'], 'cost': 132}, {'id': 'S4', 'elements': ['K', 'L', 'M'], 'cost': 231}, {'id': 'S5', 'elements': ['G', 'H', 'I'], 'cost': 162}, {'id': 'S6', 'elements': ['M', 'N', 'O'], 'cost': 231}, {'id': 'S7', 'elements': ['A', 'B', 'D'], 'cost': 66}, {'id': 'S8', 'elements': ['A', 'B', 'C'], 'cost': 291}, {'id': 'S9', 'elements': ['I', 'M'], 'cost': 68}, {'id': 'S10', 'elements': ['L', 'M'], 'cost': 148}, {'id': 'S11', 'elements': ['F', 'G'], 'cost': 152}, {'id': 'S12', 'elements': ['A', 'B', 'C', 'D'], 'cost': 44}, {'id': 'S13', 'elements': ['J', 'L'], 'cost': 118}, {'id': 'S14', 'elements': ['H', 'I', 'J', 'K'], 'cost': 164}, {'id': 'S15', 'elements': ['A', 'C'], 'cost': 88}, {'id': 'S16', 'elements': ['H', 'J', 'L'], 'cost': 90}, {'id': 'S17', 'elements': ['H', 'J', 'K', 'L'], 'cost': 316}, {'id': 'S18', 'elements': ['F', 'G', 'H', 'J'], 'cost': 388}, {'id': 'S19', 'elements': ['K', 'L', 'M', 'O'], 'cost': 168}, {'id': 'S20', 'elements': ['B', 'D'], 'cost': 136}, {'id': 'S21', 'elements': ['D', 'E', 'F', 'G'], 'cost': 236}, {'id': 'S22', 'elements': ['I', 'J', 'K'], 'cost': 72}, {'id': 'S23', 'elements': ['C', 'D', 'F'], 'cost': 69}, {'id': 'S24', 'elements': ['F', 'G', 'J'], 'cost': 240}, {'id': 'S25', 'elements': ['F', 'H'], 'cost': 50}, {'id': 'S26', 'elements': ['F', 'G', 'I', 'J'], 'cost': 20}, {'id': 'S27', 'elements': ['H', 'J'], 'cost': 146}, {'id': 'S28', 'elements': ['F', 'G', 'H'], 'cost': 171}, {'id': 'S29', 'elements': ['E', 'H', 'I'], 'cost': 117}, {'id': 'S30', 'elements': ['L', 'M', 'N', 'O'], 'cost': 364}, {'id': 'S31', 'elements': ['E', 'F', 'G'], 'cost': 81}, {'id': 'S32', 'elements': ['D', 'E', 'G', 'H'], 'cost': 52}, {'id': 'S33', 'elements': ['N', 'O'], 'cost': 72}, {'id': 'S34', 'elements': ['M', 'N'], 'cost': 148}, {'id': 'S35', 'elements': ['J', 'K', 'L'], 'cost': 297}, {'id': 'S36', 'elements': ['I', 'J'], 'cost': 96}, {'id': 'S37', 'elements': ['D', 'G', 'I'], 'cost': 282}, {'id': 'S38', 'elements': ['H', 'I'], 'cost': 18}, {'id': 'S39', 'elements': ['E', 'F', 'G', 'H'], 'cost': 148}, {'id': 'S40', 'elements': ['H', 'K', 'L'], 'cost': 264}, {'id': 'S41', 'elements': ['A'], 'cost': 10000}, {'id': 'S42', 'elements': ['B'], 'cost': 10000}, {'id': 'S43', 'elements': ['C'], 'cost': 10000}, {'id': 'S44', 'elements': ['D'], 'cost': 10000}, {'id': 'S45', 'elements': ['E'], 'cost': 10000}, {'id': 'S46', 'elements': ['F'], 'cost': 10000}, {'id': 'S47', 'elements': ['G'], 'cost': 10000}, {'id': 'S48', 'elements': ['H'], 'cost': 10000}, {'id': 'S49', 'elements': ['I'], 'cost': 10000}, {'id': 'S50', 'elements': ['J'], 'cost': 10000}, {'id': 'S51', 'elements': ['K'], 'cost': 10000}, {'id': 'S52', 'elements': ['L'], 'cost': 10000}, {'id': 'S53', 'elements': ['M'], 'cost': 10000}, {'id': 'S54', 'elements': ['N'], 'cost': 10000}, {'id': 'S55', 'elements': ['O'], 'cost': 10000}]}","['S10', 'S12', 'S22', 'S33', 'S39']",8,csv,names
SPP,SPP,"Back at the office the plan was sketched on a whiteboard: choose some clusters to host the company’s services, making sure each service ends up on exactly one chosen cluster — nothing missing and nothing running in two places. What makes one plan preferable is the final hosting charge; it’s calculated by summing the cost of every cluster we include, and the smallest sum wins. The concrete list of services, available clusters, and their costs appears below.

# total_services=13
# total_clusters=45
cluster_id,cluster_cost,services
S1,177,4 6 9
S2,89,5
S3,188,8 9 11 12
S4,14,5 6
S5,110,4 5
S6,228,9 10 11
S7,88,7 10
S8,36,4 6 7
S9,147,0 1 2
S10,27,10 11 12
S11,176,10 11
S12,194,1 2
S13,82,10 12
S14,231,1 2 4
S15,255,4 5 6
S16,28,1 3 4 5
S17,192,9 11 12
S18,123,1 3 4
S19,46,9 10
S20,84,9 10 12
S21,51,2
S22,40,2 6
S23,249,5 6 7
S24,31,11
S25,15,3
S26,63,1 2 3
S27,188,3 4
S28,136,9 10 11 12
S29,24,8 9 10
S30,222,8 9 11
S31,20,1 2 3 5
S32,201,0 1 3
S33,10000,0
S34,10000,1
S35,10000,2
S36,10000,3
S37,10000,4
S38,10000,5
S39,10000,6
S40,10000,7
S41,10000,8
S42,10000,9
S43,10000,10
S44,10000,11
S45,10000,12

If you want the plan in a neat, machine-friendly shape, just return a small JSON object like this:

{
  ""solution"": [""cluster_id"", ...]
}

""solution"" should list the clusters you picked to host each service — each entry is a cluster identifier placeholder (so in the real reply you'll replace ""cluster_id"" with the actual IDs from the instance). This is just a sketch of the shape I expect, not the actual answer.

Please make sure to use the exact identifiers from the input — do not rename them or invent new ones. For example:
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 13, 'num_sets': 45, 'density': 0.1641025641025641, 'sets': [{'id': 1, 'elements': [5, 7, 10], 'cost': 177}, {'id': 2, 'elements': [6], 'cost': 89}, {'id': 3, 'elements': [9, 10, 12, 13], 'cost': 188}, {'id': 4, 'elements': [6, 7], 'cost': 14}, {'id': 5, 'elements': [5, 6], 'cost': 110}, {'id': 6, 'elements': [10, 11, 12], 'cost': 228}, {'id': 7, 'elements': [8, 11], 'cost': 88}, {'id': 8, 'elements': [5, 7, 8], 'cost': 36}, {'id': 9, 'elements': [1, 2, 3], 'cost': 147}, {'id': 10, 'elements': [11, 12, 13], 'cost': 27}, {'id': 11, 'elements': [11, 12], 'cost': 176}, {'id': 12, 'elements': [2, 3], 'cost': 194}, {'id': 13, 'elements': [11, 13], 'cost': 82}, {'id': 14, 'elements': [2, 3, 5], 'cost': 231}, {'id': 15, 'elements': [5, 6, 7], 'cost': 255}, {'id': 16, 'elements': [2, 4, 5, 6], 'cost': 28}, {'id': 17, 'elements': [10, 12, 13], 'cost': 192}, {'id': 18, 'elements': [2, 4, 5], 'cost': 123}, {'id': 19, 'elements': [10, 11], 'cost': 46}, {'id': 20, 'elements': [10, 11, 13], 'cost': 84}, {'id': 21, 'elements': [3], 'cost': 51}, {'id': 22, 'elements': [3, 7], 'cost': 40}, {'id': 23, 'elements': [6, 7, 8], 'cost': 249}, {'id': 24, 'elements': [12], 'cost': 31}, {'id': 25, 'elements': [4], 'cost': 15}, {'id': 26, 'elements': [2, 3, 4], 'cost': 63}, {'id': 27, 'elements': [4, 5], 'cost': 188}, {'id': 28, 'elements': [10, 11, 12, 13], 'cost': 136}, {'id': 29, 'elements': [9, 10, 11], 'cost': 24}, {'id': 30, 'elements': [9, 10, 12], 'cost': 222}, {'id': 31, 'elements': [2, 3, 4, 6], 'cost': 20}, {'id': 32, 'elements': [1, 2, 4], 'cost': 201}, {'id': 33, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 34, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 35, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 36, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 37, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 38, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 39, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 40, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 41, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 42, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 43, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 44, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 45, 'elements': [13], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0008_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0008_bag_stats.png'}","[2, 8, 9, 13, 25, 30]",591.0,"{'num_elements': 13, 'num_sets': 45, 'sets': [{'id': 'S1', 'elements': [4, 6, 9], 'cost': 177}, {'id': 'S2', 'elements': [5], 'cost': 89}, {'id': 'S3', 'elements': [8, 9, 11, 12], 'cost': 188}, {'id': 'S4', 'elements': [5, 6], 'cost': 14}, {'id': 'S5', 'elements': [4, 5], 'cost': 110}, {'id': 'S6', 'elements': [9, 10, 11], 'cost': 228}, {'id': 'S7', 'elements': [7, 10], 'cost': 88}, {'id': 'S8', 'elements': [4, 6, 7], 'cost': 36}, {'id': 'S9', 'elements': [0, 1, 2], 'cost': 147}, {'id': 'S10', 'elements': [10, 11, 12], 'cost': 27}, {'id': 'S11', 'elements': [10, 11], 'cost': 176}, {'id': 'S12', 'elements': [1, 2], 'cost': 194}, {'id': 'S13', 'elements': [10, 12], 'cost': 82}, {'id': 'S14', 'elements': [1, 2, 4], 'cost': 231}, {'id': 'S15', 'elements': [4, 5, 6], 'cost': 255}, {'id': 'S16', 'elements': [1, 3, 4, 5], 'cost': 28}, {'id': 'S17', 'elements': [9, 11, 12], 'cost': 192}, {'id': 'S18', 'elements': [1, 3, 4], 'cost': 123}, {'id': 'S19', 'elements': [9, 10], 'cost': 46}, {'id': 'S20', 'elements': [9, 10, 12], 'cost': 84}, {'id': 'S21', 'elements': [2], 'cost': 51}, {'id': 'S22', 'elements': [2, 6], 'cost': 40}, {'id': 'S23', 'elements': [5, 6, 7], 'cost': 249}, {'id': 'S24', 'elements': [11], 'cost': 31}, {'id': 'S25', 'elements': [3], 'cost': 15}, {'id': 'S26', 'elements': [1, 2, 3], 'cost': 63}, {'id': 'S27', 'elements': [3, 4], 'cost': 188}, {'id': 'S28', 'elements': [9, 10, 11, 12], 'cost': 136}, {'id': 'S29', 'elements': [8, 9, 10], 'cost': 24}, {'id': 'S30', 'elements': [8, 9, 11], 'cost': 222}, {'id': 'S31', 'elements': [1, 2, 3, 5], 'cost': 20}, {'id': 'S32', 'elements': [0, 1, 3], 'cost': 201}, {'id': 'S33', 'elements': [0], 'cost': 10000}, {'id': 'S34', 'elements': [1], 'cost': 10000}, {'id': 'S35', 'elements': [2], 'cost': 10000}, {'id': 'S36', 'elements': [3], 'cost': 10000}, {'id': 'S37', 'elements': [4], 'cost': 10000}, {'id': 'S38', 'elements': [5], 'cost': 10000}, {'id': 'S39', 'elements': [6], 'cost': 10000}, {'id': 'S40', 'elements': [7], 'cost': 10000}, {'id': 'S41', 'elements': [8], 'cost': 10000}, {'id': 'S42', 'elements': [9], 'cost': 10000}, {'id': 'S43', 'elements': [10], 'cost': 10000}, {'id': 'S44', 'elements': [11], 'cost': 10000}, {'id': 'S45', 'elements': [12], 'cost': 10000}]}","['S2', 'S8', 'S9', 'S13', 'S25', 'S30']",9,csv,0
SPP,SPP,"There’s a community event coming up where every little program needs its own covered spot, so the task is to choose a set of tents to rent that will host all activities. Each activity must be hosted in one, and only one, of the tents we pick (no splitting and no leaving anyone out), and the goal is to make the rental bill as low as possible by adding up the costs of the tents selected. The exact tent options and the activity list are shown below.

# total_activities=13
# candidate_tents_count=61
tent_id,rental_cost,activities
S1,100,I
S2,114,D E F
S3,80,J K L M
S4,70,D
S5,2,L M
S6,204,E G H
S7,48,E G H K
S8,67,F
S9,150,F G
S10,152,B C
S11,123,E F H
S12,40,E F G H
S13,240,A B C D
S14,54,B E
S15,156,J L
S16,92,E
S17,78,D F
S18,116,A C
S19,54,H I K
S20,34,C D
S21,166,G H
S22,6,I K
S23,172,B D E F
S24,112,F J
S25,138,F G H
S26,76,I J
S27,162,G H I
S28,186,J L M
S29,200,A B
S30,105,B D F
S31,304,B C D E
S32,73,K
S33,192,J K L
S34,87,A B C
S35,168,B C E F
S36,19,C
S37,255,H J K
S38,92,J K
S39,222,D F G
S40,156,J M
S41,146,K M
S42,62,G I
S43,24,E H
S44,102,J K M
S45,98,C E
S46,148,E G
S47,14,A
S48,24,B D
S49,10000,A
S50,10000,B
S51,10000,C
S52,10000,D
S53,10000,E
S54,10000,F
S55,10000,G
S56,10000,H
S57,10000,I
S58,10000,J
S59,10000,K
S60,10000,L
S61,10000,M

Also, when you send back your choice, please use this simple JSON layout so it's easy to read by both people and tools:

{
  ""solution"": [""tent_id"", ...]
}

This is just a sketch of the shape I expect:
- ""solution"" is the list of the tents you want to rent (one entry per tent).
- The ""tent_id"" placeholder is where you'll put the actual tent identifiers from the instance (replace the placeholder with the real IDs).
- The JSON itself is only an outline — fill in the real tent IDs when you submit the answer.

Please make sure you use the exact identifiers from the instance input — do not rename them and do not invent new labels.
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 13, 'num_sets': 61, 'density': 0.1626733921815889, 'sets': [{'id': 1, 'elements': [9], 'cost': 100}, {'id': 2, 'elements': [4, 5, 6], 'cost': 114}, {'id': 3, 'elements': [10, 11, 12, 13], 'cost': 80}, {'id': 4, 'elements': [4], 'cost': 70}, {'id': 5, 'elements': [12, 13], 'cost': 2}, {'id': 6, 'elements': [5, 7, 8], 'cost': 204}, {'id': 7, 'elements': [5, 7, 8, 11], 'cost': 48}, {'id': 8, 'elements': [6], 'cost': 67}, {'id': 9, 'elements': [6, 7], 'cost': 150}, {'id': 10, 'elements': [2, 3], 'cost': 152}, {'id': 11, 'elements': [5, 6, 8], 'cost': 123}, {'id': 12, 'elements': [5, 6, 7, 8], 'cost': 40}, {'id': 13, 'elements': [1, 2, 3, 4], 'cost': 240}, {'id': 14, 'elements': [2, 5], 'cost': 54}, {'id': 15, 'elements': [10, 12], 'cost': 156}, {'id': 16, 'elements': [5], 'cost': 92}, {'id': 17, 'elements': [4, 6], 'cost': 78}, {'id': 18, 'elements': [1, 3], 'cost': 116}, {'id': 19, 'elements': [8, 9, 11], 'cost': 54}, {'id': 20, 'elements': [3, 4], 'cost': 34}, {'id': 21, 'elements': [7, 8], 'cost': 166}, {'id': 22, 'elements': [9, 11], 'cost': 6}, {'id': 23, 'elements': [2, 4, 5, 6], 'cost': 172}, {'id': 24, 'elements': [6, 10], 'cost': 112}, {'id': 25, 'elements': [6, 7, 8], 'cost': 138}, {'id': 26, 'elements': [9, 10], 'cost': 76}, {'id': 27, 'elements': [7, 8, 9], 'cost': 162}, {'id': 28, 'elements': [10, 12, 13], 'cost': 186}, {'id': 29, 'elements': [1, 2], 'cost': 200}, {'id': 30, 'elements': [2, 4, 6], 'cost': 105}, {'id': 31, 'elements': [2, 3, 4, 5], 'cost': 304}, {'id': 32, 'elements': [11], 'cost': 73}, {'id': 33, 'elements': [10, 11, 12], 'cost': 192}, {'id': 34, 'elements': [1, 2, 3], 'cost': 87}, {'id': 35, 'elements': [2, 3, 5, 6], 'cost': 168}, {'id': 36, 'elements': [3], 'cost': 19}, {'id': 37, 'elements': [8, 10, 11], 'cost': 255}, {'id': 38, 'elements': [10, 11], 'cost': 92}, {'id': 39, 'elements': [4, 6, 7], 'cost': 222}, {'id': 40, 'elements': [10, 13], 'cost': 156}, {'id': 41, 'elements': [11, 13], 'cost': 146}, {'id': 42, 'elements': [7, 9], 'cost': 62}, {'id': 43, 'elements': [5, 8], 'cost': 24}, {'id': 44, 'elements': [10, 11, 13], 'cost': 102}, {'id': 45, 'elements': [3, 5], 'cost': 98}, {'id': 46, 'elements': [5, 7], 'cost': 148}, {'id': 47, 'elements': [1], 'cost': 14}, {'id': 48, 'elements': [2, 4], 'cost': 24}, {'id': 49, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [13], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0009_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0009_bag_stats.png'}","[5, 12, 26, 32, 36, 47, 48]",248.0,"{'num_elements': 13, 'num_sets': 61, 'sets': [{'id': 'S1', 'elements': ['I'], 'cost': 100}, {'id': 'S2', 'elements': ['D', 'E', 'F'], 'cost': 114}, {'id': 'S3', 'elements': ['J', 'K', 'L', 'M'], 'cost': 80}, {'id': 'S4', 'elements': ['D'], 'cost': 70}, {'id': 'S5', 'elements': ['L', 'M'], 'cost': 2}, {'id': 'S6', 'elements': ['E', 'G', 'H'], 'cost': 204}, {'id': 'S7', 'elements': ['E', 'G', 'H', 'K'], 'cost': 48}, {'id': 'S8', 'elements': ['F'], 'cost': 67}, {'id': 'S9', 'elements': ['F', 'G'], 'cost': 150}, {'id': 'S10', 'elements': ['B', 'C'], 'cost': 152}, {'id': 'S11', 'elements': ['E', 'F', 'H'], 'cost': 123}, {'id': 'S12', 'elements': ['E', 'F', 'G', 'H'], 'cost': 40}, {'id': 'S13', 'elements': ['A', 'B', 'C', 'D'], 'cost': 240}, {'id': 'S14', 'elements': ['B', 'E'], 'cost': 54}, {'id': 'S15', 'elements': ['J', 'L'], 'cost': 156}, {'id': 'S16', 'elements': ['E'], 'cost': 92}, {'id': 'S17', 'elements': ['D', 'F'], 'cost': 78}, {'id': 'S18', 'elements': ['A', 'C'], 'cost': 116}, {'id': 'S19', 'elements': ['H', 'I', 'K'], 'cost': 54}, {'id': 'S20', 'elements': ['C', 'D'], 'cost': 34}, {'id': 'S21', 'elements': ['G', 'H'], 'cost': 166}, {'id': 'S22', 'elements': ['I', 'K'], 'cost': 6}, {'id': 'S23', 'elements': ['B', 'D', 'E', 'F'], 'cost': 172}, {'id': 'S24', 'elements': ['F', 'J'], 'cost': 112}, {'id': 'S25', 'elements': ['F', 'G', 'H'], 'cost': 138}, {'id': 'S26', 'elements': ['I', 'J'], 'cost': 76}, {'id': 'S27', 'elements': ['G', 'H', 'I'], 'cost': 162}, {'id': 'S28', 'elements': ['J', 'L', 'M'], 'cost': 186}, {'id': 'S29', 'elements': ['A', 'B'], 'cost': 200}, {'id': 'S30', 'elements': ['B', 'D', 'F'], 'cost': 105}, {'id': 'S31', 'elements': ['B', 'C', 'D', 'E'], 'cost': 304}, {'id': 'S32', 'elements': ['K'], 'cost': 73}, {'id': 'S33', 'elements': ['J', 'K', 'L'], 'cost': 192}, {'id': 'S34', 'elements': ['A', 'B', 'C'], 'cost': 87}, {'id': 'S35', 'elements': ['B', 'C', 'E', 'F'], 'cost': 168}, {'id': 'S36', 'elements': ['C'], 'cost': 19}, {'id': 'S37', 'elements': ['H', 'J', 'K'], 'cost': 255}, {'id': 'S38', 'elements': ['J', 'K'], 'cost': 92}, {'id': 'S39', 'elements': ['D', 'F', 'G'], 'cost': 222}, {'id': 'S40', 'elements': ['J', 'M'], 'cost': 156}, {'id': 'S41', 'elements': ['K', 'M'], 'cost': 146}, {'id': 'S42', 'elements': ['G', 'I'], 'cost': 62}, {'id': 'S43', 'elements': ['E', 'H'], 'cost': 24}, {'id': 'S44', 'elements': ['J', 'K', 'M'], 'cost': 102}, {'id': 'S45', 'elements': ['C', 'E'], 'cost': 98}, {'id': 'S46', 'elements': ['E', 'G'], 'cost': 148}, {'id': 'S47', 'elements': ['A'], 'cost': 14}, {'id': 'S48', 'elements': ['B', 'D'], 'cost': 24}, {'id': 'S49', 'elements': ['A'], 'cost': 10000}, {'id': 'S50', 'elements': ['B'], 'cost': 10000}, {'id': 'S51', 'elements': ['C'], 'cost': 10000}, {'id': 'S52', 'elements': ['D'], 'cost': 10000}, {'id': 'S53', 'elements': ['E'], 'cost': 10000}, {'id': 'S54', 'elements': ['F'], 'cost': 10000}, {'id': 'S55', 'elements': ['G'], 'cost': 10000}, {'id': 'S56', 'elements': ['H'], 'cost': 10000}, {'id': 'S57', 'elements': ['I'], 'cost': 10000}, {'id': 'S58', 'elements': ['J'], 'cost': 10000}, {'id': 'S59', 'elements': ['K'], 'cost': 10000}, {'id': 'S60', 'elements': ['L'], 'cost': 10000}, {'id': 'S61', 'elements': ['M'], 'cost': 10000}]}","['S5', 'S12', 'S26', 'S32', 'S36', 'S47', 'S48']",10,csv,names
SPP,SPP,"A hospital manager has to decide which combination of care teams to call in so that each patient is assigned to exactly one team. What makes one combination better is its total cost — you calculate it by summing the staffing cost of all the teams chosen, and the goal is to keep that sum as small as possible. The constraint is practical and simple: every patient must be covered, and no patient can be assigned to more than one team. The concrete details are provided below.

{
  ""total_patients"": 15,
  ""num_candidate_teams"": 63,
  ""sets"": [
    {
      ""team_id"": ""S1"",
      ""covered_patient_ids"": [
        7,
        8,
        9
      ],
      ""team_cost"": 234
    },
    {
      ""team_id"": ""S2"",
      ""covered_patient_ids"": [
        1,
        3,
        4,
        6
      ],
      ""team_cost"": 284
    },
    {
      ""team_id"": ""S3"",
      ""covered_patient_ids"": [
        9,
        10,
        12,
        13
      ],
      ""team_cost"": 204
    },
    {
      ""team_id"": ""S4"",
      ""covered_patient_ids"": [
        10,
        11,
        14
      ],
      ""team_cost"": 36
    },
    {
      ""team_id"": ""S5"",
      ""covered_patient_ids"": [
        10,
        11
      ],
      ""team_cost"": 26
    },
    {
      ""team_id"": ""S6"",
      ""covered_patient_ids"": [
        4,
        6
      ],
      ""team_cost"": 122
    },
    {
      ""team_id"": ""S7"",
      ""covered_patient_ids"": [
        3,
        5,
        6,
        7
      ],
      ""team_cost"": 332
    },
    {
      ""team_id"": ""S8"",
      ""covered_patient_ids"": [
        1,
        2,
        3
      ],
      ""team_cost"": 18
    },
    {
      ""team_id"": ""S9"",
      ""covered_patient_ids"": [
        7,
        10
      ],
      ""team_cost"": 146
    },
    {
      ""team_id"": ""S10"",
      ""covered_patient_ids"": [
        11,
        14
      ],
      ""team_cost"": 172
    },
    {
      ""team_id"": ""S11"",
      ""covered_patient_ids"": [
        7,
        9,
        10
      ],
      ""team_cost"": 24
    },
    {
      ""team_id"": ""S12"",
      ""covered_patient_ids"": [
        12,
        13,
        14,
        15
      ],
      ""team_cost"": 296
    },
    {
      ""team_id"": ""S13"",
      ""covered_patient_ids"": [
        5,
        7,
        8,
        10
      ],
      ""team_cost"": 296
    },
    {
      ""team_id"": ""S14"",
      ""covered_patient_ids"": [
        11,
        13,
        14,
        15
      ],
      ""team_cost"": 232
    },
    {
      ""team_id"": ""S15"",
      ""covered_patient_ids"": [
        5,
        8
      ],
      ""team_cost"": 52
    },
    {
      ""team_id"": ""S16"",
      ""covered_patient_ids"": [
        1,
        2,
        3,
        4
      ],
      ""team_cost"": 236
    },
    {
      ""team_id"": ""S17"",
      ""covered_patient_ids"": [
        3,
        4,
        6
      ],
      ""team_cost"": 39
    },
    {
      ""team_id"": ""S18"",
      ""covered_patient_ids"": [
        9,
        10,
        11,
        12
      ],
      ""team_cost"": 212
    },
    {
      ""team_id"": ""S19"",
      ""covered_patient_ids"": [
        11,
        12
      ],
      ""team_cost"": 10
    },
    {
      ""team_id"": ""S20"",
      ""covered_patient_ids"": [
        2,
        6,
        7
      ],
      ""team_cost"": 117
    },
    {
      ""team_id"": ""S21"",
      ""covered_patient_ids"": [
        14,
        15
      ],
      ""team_cost"": 26
    },
    {
      ""team_id"": ""S22"",
      ""covered_patient_ids"": [
        2,
        4,
        6,
        7
      ],
      ""team_cost"": 140
    },
    {
      ""team_id"": ""S23"",
      ""covered_patient_ids"": [
        8,
        9,
        11,
        12
      ],
      ""team_cost"": 380
    },
    {
      ""team_id"": ""S24"",
      ""covered_patient_ids"": [
        8,
        10,
        11
      ],
      ""team_cost"": 63
    },
    {
      ""team_id"": ""S25"",
      ""covered_patient_ids"": [
        10,
        12
      ],
      ""team_cost"": 136
    },
    {
      ""team_id"": ""S26"",
      ""covered_patient_ids"": [
        2,
        3
      ],
      ""team_cost"": 72
    },
    {
      ""team_id"": ""S27"",
      ""covered_patient_ids"": [
        6,
        8,
        9
      ],
      ""team_cost"": 12
    },
    {
      ""team_id"": ""S28"",
      ""covered_patient_ids"": [
        6,
        8
      ],
      ""team_cost"": 122
    },
    {
      ""team_id"": ""S29"",
      ""covered_patient_ids"": [
        5,
        6,
        8
      ],
      ""team_cost"": 267
    },
    {
      ""team_id"": ""S30"",
      ""covered_patient_ids"": [
        1,
        2
      ],
      ""team_cost"": 148
    },
    {
      ""team_id"": ""S31"",
      ""covered_patient_ids"": [
        2,
        4,
        5,
        6
      ],
      ""team_cost"": 212
    },
    {
      ""team_id"": ""S32"",
      ""covered_patient_ids"": [
        10,
        14,
        15
      ],
      ""team_cost"": 51
    },
    {
      ""team_id"": ""S33"",
      ""covered_patient_ids"": [
        9,
        10,
        11
      ],
      ""team_cost"": 120
    },
    {
      ""team_id"": ""S34"",
      ""covered_patient_ids"": [
        10,
        11,
        12,
        13
      ],
      ""team_cost"": 132
    },
    {
      ""team_id"": ""S35"",
      ""covered_patient_ids"": [
        6,
        7
      ],
      ""team_cost"": 102
    },
    {
      ""team_id"": ""S36"",
      ""covered_patient_ids"": [
        10,
        11,
        12
      ],
      ""team_cost"": 186
    },
    {
      ""team_id"": ""S37"",
      ""covered_patient_ids"": [
        12,
        14
      ],
      ""team_cost"": 50
    },
    {
      ""team_id"": ""S38"",
      ""covered_patient_ids"": [
        4,
        5,
        6
      ],
      ""team_cost"": 15
    },
    {
      ""team_id"": ""S39"",
      ""covered_patient_ids"": [
        10,
        12,
        13,
        14
      ],
      ""team_cost"": 172
    },
    {
      ""team_id"": ""S40"",
      ""covered_patient_ids"": [
        3,
        4,
        5
      ],
      ""team_cost"": 129
    },
    {
      ""team_id"": ""S41"",
      ""covered_patient_ids"": [
        9,
        10,
        12
      ],
      ""team_cost"": 246
    },
    {
      ""team_id"": ""S42"",
      ""covered_patient_ids"": [
        8,
        10,
        12
      ],
      ""team_cost"": 264
    },
    {
      ""team_id"": ""S43"",
      ""covered_patient_ids"": [
        3,
        5,
        7
      ],
      ""team_cost"": 78
    },
    {
      ""team_id"": ""S44"",
      ""covered_patient_ids"": [
        8,
        12,
        13
      ],
      ""team_cost"": 276
    },
    {
      ""team_id"": ""S45"",
      ""covered_patient_ids"": [
        9,
        11
      ],
      ""team_cost"": 54
    },
    {
      ""team_id"": ""S46"",
      ""covered_patient_ids"": [
        1,
        3,
        4,
        5
      ],
      ""team_cost"": 68
    },
    {
      ""team_id"": ""S47"",
      ""covered_patient_ids"": [
        8,
        9
      ],
      ""team_cost"": 4
    },
    {
      ""team_id"": ""S48"",
      ""covered_patient_ids"": [
        11,
        12,
        13,
        14
      ],
      ""team_cost"": 128
    },
    {
      ""team_id"": ""S49"",
      ""covered_patient_ids"": [
        1
      ],
      ""team_cost"": 10000
    },
    {
      ""team_id"": ""S50"",
      ""covered_patient_ids"": [
        2
      ],
      ""team_cost"": 10000
    },
    {
      ""team_id"": ""S51"",
      ""covered_patient_ids"": [
        3
      ],
      ""team_cost"": 10000
    },
    {
      ""team_id"": ""S52"",
      ""covered_patient_ids"": [
        4
      ],
      ""team_cost"": 10000
    },
    {
      ""team_id"": ""S53"",
      ""covered_patient_ids"": [
        5
      ],
      ""team_cost"": 10000
    },
    {
      ""team_id"": ""S54"",
      ""covered_patient_ids"": [
        6
      ],
      ""team_cost"": 10000
    },
    {
      ""team_id"": ""S55"",
      ""covered_patient_ids"": [
        7
      ],
      ""team_cost"": 10000
    },
    {
      ""team_id"": ""S56"",
      ""covered_patient_ids"": [
        8
      ],
      ""team_cost"": 10000
    },
    {
      ""team_id"": ""S57"",
      ""covered_patient_ids"": [
        9
      ],
      ""team_cost"": 10000
    },
    {
      ""team_id"": ""S58"",
      ""covered_patient_ids"": [
        10
      ],
      ""team_cost"": 10000
    },
    {
      ""team_id"": ""S59"",
      ""covered_patient_ids"": [
        11
      ],
      ""team_cost"": 10000
    },
    {
      ""team_id"": ""S60"",
      ""covered_patient_ids"": [
        12
      ],
      ""team_cost"": 10000
    },
    {
      ""team_id"": ""S61"",
      ""covered_patient_ids"": [
        13
      ],
      ""team_cost"": 10000
    },
    {
      ""team_id"": ""S62"",
      ""covered_patient_ids"": [
        14
      ],
      ""team_cost"": 10000
    },
    {
      ""team_id"": ""S63"",
      ""covered_patient_ids"": [
        15
      ],
      ""team_cost"": 10000
    }
  ]
}

Also, when you reply, please put the chosen teams into a little JSON snippet like this so it's easy to read and process:

{
  ""solution"": [""team_id"", ...]
}

""solution"" is the top-level field holding the list of care teams to call in. Each item in that list is a team identifier (I'm showing the placeholder ""team_id"" here to match the hospital story) — in your actual submission replace those placeholders with the exact identifiers from the instance. This JSON is just a sketch of the shape I expect, not the final answer itself.

Please make sure you use the identifiers exactly as they appear in the instance input — no renaming and no new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'problem_type': 'SPP', 'num_elements': 15, 'num_sets': 63, 'density': 0.16825396825396827, 'sets': [{'id': 1, 'elements': [7, 8, 9], 'cost': 234}, {'id': 2, 'elements': [1, 3, 4, 6], 'cost': 284}, {'id': 3, 'elements': [9, 10, 12, 13], 'cost': 204}, {'id': 4, 'elements': [10, 11, 14], 'cost': 36}, {'id': 5, 'elements': [10, 11], 'cost': 26}, {'id': 6, 'elements': [4, 6], 'cost': 122}, {'id': 7, 'elements': [3, 5, 6, 7], 'cost': 332}, {'id': 8, 'elements': [1, 2, 3], 'cost': 18}, {'id': 9, 'elements': [7, 10], 'cost': 146}, {'id': 10, 'elements': [11, 14], 'cost': 172}, {'id': 11, 'elements': [7, 9, 10], 'cost': 24}, {'id': 12, 'elements': [12, 13, 14, 15], 'cost': 296}, {'id': 13, 'elements': [5, 7, 8, 10], 'cost': 296}, {'id': 14, 'elements': [11, 13, 14, 15], 'cost': 232}, {'id': 15, 'elements': [5, 8], 'cost': 52}, {'id': 16, 'elements': [1, 2, 3, 4], 'cost': 236}, {'id': 17, 'elements': [3, 4, 6], 'cost': 39}, {'id': 18, 'elements': [9, 10, 11, 12], 'cost': 212}, {'id': 19, 'elements': [11, 12], 'cost': 10}, {'id': 20, 'elements': [2, 6, 7], 'cost': 117}, {'id': 21, 'elements': [14, 15], 'cost': 26}, {'id': 22, 'elements': [2, 4, 6, 7], 'cost': 140}, {'id': 23, 'elements': [8, 9, 11, 12], 'cost': 380}, {'id': 24, 'elements': [8, 10, 11], 'cost': 63}, {'id': 25, 'elements': [10, 12], 'cost': 136}, {'id': 26, 'elements': [2, 3], 'cost': 72}, {'id': 27, 'elements': [6, 8, 9], 'cost': 12}, {'id': 28, 'elements': [6, 8], 'cost': 122}, {'id': 29, 'elements': [5, 6, 8], 'cost': 267}, {'id': 30, 'elements': [1, 2], 'cost': 148}, {'id': 31, 'elements': [2, 4, 5, 6], 'cost': 212}, {'id': 32, 'elements': [10, 14, 15], 'cost': 51}, {'id': 33, 'elements': [9, 10, 11], 'cost': 120}, {'id': 34, 'elements': [10, 11, 12, 13], 'cost': 132}, {'id': 35, 'elements': [6, 7], 'cost': 102}, {'id': 36, 'elements': [10, 11, 12], 'cost': 186}, {'id': 37, 'elements': [12, 14], 'cost': 50}, {'id': 38, 'elements': [4, 5, 6], 'cost': 15}, {'id': 39, 'elements': [10, 12, 13, 14], 'cost': 172}, {'id': 40, 'elements': [3, 4, 5], 'cost': 129}, {'id': 41, 'elements': [9, 10, 12], 'cost': 246}, {'id': 42, 'elements': [8, 10, 12], 'cost': 264}, {'id': 43, 'elements': [3, 5, 7], 'cost': 78}, {'id': 44, 'elements': [8, 12, 13], 'cost': 276}, {'id': 45, 'elements': [9, 11], 'cost': 54}, {'id': 46, 'elements': [1, 3, 4, 5], 'cost': 68}, {'id': 47, 'elements': [8, 9], 'cost': 4}, {'id': 48, 'elements': [11, 12, 13, 14], 'cost': 128}, {'id': 49, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [15], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0010_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0010_bag_stats.png'}","[20, 21, 34, 46, 47]",347.0,"{'num_elements': 15, 'num_sets': 63, 'sets': [{'id': 'S1', 'elements': [7, 8, 9], 'cost': 234}, {'id': 'S2', 'elements': [1, 3, 4, 6], 'cost': 284}, {'id': 'S3', 'elements': [9, 10, 12, 13], 'cost': 204}, {'id': 'S4', 'elements': [10, 11, 14], 'cost': 36}, {'id': 'S5', 'elements': [10, 11], 'cost': 26}, {'id': 'S6', 'elements': [4, 6], 'cost': 122}, {'id': 'S7', 'elements': [3, 5, 6, 7], 'cost': 332}, {'id': 'S8', 'elements': [1, 2, 3], 'cost': 18}, {'id': 'S9', 'elements': [7, 10], 'cost': 146}, {'id': 'S10', 'elements': [11, 14], 'cost': 172}, {'id': 'S11', 'elements': [7, 9, 10], 'cost': 24}, {'id': 'S12', 'elements': [12, 13, 14, 15], 'cost': 296}, {'id': 'S13', 'elements': [5, 7, 8, 10], 'cost': 296}, {'id': 'S14', 'elements': [11, 13, 14, 15], 'cost': 232}, {'id': 'S15', 'elements': [5, 8], 'cost': 52}, {'id': 'S16', 'elements': [1, 2, 3, 4], 'cost': 236}, {'id': 'S17', 'elements': [3, 4, 6], 'cost': 39}, {'id': 'S18', 'elements': [9, 10, 11, 12], 'cost': 212}, {'id': 'S19', 'elements': [11, 12], 'cost': 10}, {'id': 'S20', 'elements': [2, 6, 7], 'cost': 117}, {'id': 'S21', 'elements': [14, 15], 'cost': 26}, {'id': 'S22', 'elements': [2, 4, 6, 7], 'cost': 140}, {'id': 'S23', 'elements': [8, 9, 11, 12], 'cost': 380}, {'id': 'S24', 'elements': [8, 10, 11], 'cost': 63}, {'id': 'S25', 'elements': [10, 12], 'cost': 136}, {'id': 'S26', 'elements': [2, 3], 'cost': 72}, {'id': 'S27', 'elements': [6, 8, 9], 'cost': 12}, {'id': 'S28', 'elements': [6, 8], 'cost': 122}, {'id': 'S29', 'elements': [5, 6, 8], 'cost': 267}, {'id': 'S30', 'elements': [1, 2], 'cost': 148}, {'id': 'S31', 'elements': [2, 4, 5, 6], 'cost': 212}, {'id': 'S32', 'elements': [10, 14, 15], 'cost': 51}, {'id': 'S33', 'elements': [9, 10, 11], 'cost': 120}, {'id': 'S34', 'elements': [10, 11, 12, 13], 'cost': 132}, {'id': 'S35', 'elements': [6, 7], 'cost': 102}, {'id': 'S36', 'elements': [10, 11, 12], 'cost': 186}, {'id': 'S37', 'elements': [12, 14], 'cost': 50}, {'id': 'S38', 'elements': [4, 5, 6], 'cost': 15}, {'id': 'S39', 'elements': [10, 12, 13, 14], 'cost': 172}, {'id': 'S40', 'elements': [3, 4, 5], 'cost': 129}, {'id': 'S41', 'elements': [9, 10, 12], 'cost': 246}, {'id': 'S42', 'elements': [8, 10, 12], 'cost': 264}, {'id': 'S43', 'elements': [3, 5, 7], 'cost': 78}, {'id': 'S44', 'elements': [8, 12, 13], 'cost': 276}, {'id': 'S45', 'elements': [9, 11], 'cost': 54}, {'id': 'S46', 'elements': [1, 3, 4, 5], 'cost': 68}, {'id': 'S47', 'elements': [8, 9], 'cost': 4}, {'id': 'S48', 'elements': [11, 12, 13, 14], 'cost': 128}, {'id': 'S49', 'elements': [1], 'cost': 10000}, {'id': 'S50', 'elements': [2], 'cost': 10000}, {'id': 'S51', 'elements': [3], 'cost': 10000}, {'id': 'S52', 'elements': [4], 'cost': 10000}, {'id': 'S53', 'elements': [5], 'cost': 10000}, {'id': 'S54', 'elements': [6], 'cost': 10000}, {'id': 'S55', 'elements': [7], 'cost': 10000}, {'id': 'S56', 'elements': [8], 'cost': 10000}, {'id': 'S57', 'elements': [9], 'cost': 10000}, {'id': 'S58', 'elements': [10], 'cost': 10000}, {'id': 'S59', 'elements': [11], 'cost': 10000}, {'id': 'S60', 'elements': [12], 'cost': 10000}, {'id': 'S61', 'elements': [13], 'cost': 10000}, {'id': 'S62', 'elements': [14], 'cost': 10000}, {'id': 'S63', 'elements': [15], 'cost': 10000}]}","['S20', 'S21', 'S34', 'S46', 'S47']",11,json,1
SPP,SPP,"Someone on the crew is lining up performers and needs to decide which stages to book so every act has one assigned spot and nobody gets assigned twice. Once the stage selections are made, the total expense is just the sum of the costs for those booked stages, and the aim is to keep that sum as low as possible. The full list of acts, stage options, and their costs appears below.

# total_performers=14
# total_stage_options=60
stage_id,stage_cost,stage_performers
S1,80,A C
S2,118,K N
S3,54,I J K
S4,104,F H
S5,17,E
S6,94,A B
S7,207,J L N
S8,9,J K M
S9,6,G H I
S10,160,I K
S11,240,B C E
S12,156,B C
S13,128,D F
S14,60,H K M N
S15,372,H K L M
S16,152,B C D E
S17,26,J L
S18,228,A B C D
S19,208,C D F G
S20,162,G H J
S21,63,K L N
S22,114,J K
S23,176,G H
S24,212,J K M N
S25,261,A B D
S26,258,L M N
S27,128,G I J K
S28,216,J L M
S29,256,H I K L
S30,123,K L M
S31,104,E I
S32,176,I J K M
S33,170,L M
S34,58,F I
S35,38,G
S36,54,D E G
S37,204,J K L M
S38,28,M N
S39,56,J
S40,92,G H I J
S41,126,A C D
S42,96,F G H
S43,132,G I
S44,190,K L
S45,356,B D E F
S46,152,A B D F
S47,10000,A
S48,10000,B
S49,10000,C
S50,10000,D
S51,10000,E
S52,10000,F
S53,10000,G
S54,10000,H
S55,10000,I
S56,10000,J
S57,10000,K
S58,10000,L
S59,10000,M
S60,10000,N

When you send back which stages you want to book, just use a little JSON with this shape so it's easy to read and check:

{
  ""solution"": [""stage_id"", ...]
}

Think of it like a simple form: ""solution"" is a list of the stage IDs you choose (one per act). The ""stage_id"" placeholder stands in for whatever exact stage identifiers appear in the instance — replace those placeholders with the real IDs when you answer. This block is just a sketch of the shape I expect, not the actual selection.

Please use the identifiers exactly as they appear in the instance input — no renaming and no new labels.
- ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 14, 'num_sets': 60, 'density': 0.17023809523809524, 'sets': [{'id': 1, 'elements': [1, 3], 'cost': 80}, {'id': 2, 'elements': [11, 14], 'cost': 118}, {'id': 3, 'elements': [9, 10, 11], 'cost': 54}, {'id': 4, 'elements': [6, 8], 'cost': 104}, {'id': 5, 'elements': [5], 'cost': 17}, {'id': 6, 'elements': [1, 2], 'cost': 94}, {'id': 7, 'elements': [10, 12, 14], 'cost': 207}, {'id': 8, 'elements': [10, 11, 13], 'cost': 9}, {'id': 9, 'elements': [7, 8, 9], 'cost': 6}, {'id': 10, 'elements': [9, 11], 'cost': 160}, {'id': 11, 'elements': [2, 3, 5], 'cost': 240}, {'id': 12, 'elements': [2, 3], 'cost': 156}, {'id': 13, 'elements': [4, 6], 'cost': 128}, {'id': 14, 'elements': [8, 11, 13, 14], 'cost': 60}, {'id': 15, 'elements': [8, 11, 12, 13], 'cost': 372}, {'id': 16, 'elements': [2, 3, 4, 5], 'cost': 152}, {'id': 17, 'elements': [10, 12], 'cost': 26}, {'id': 18, 'elements': [1, 2, 3, 4], 'cost': 228}, {'id': 19, 'elements': [3, 4, 6, 7], 'cost': 208}, {'id': 20, 'elements': [7, 8, 10], 'cost': 162}, {'id': 21, 'elements': [11, 12, 14], 'cost': 63}, {'id': 22, 'elements': [10, 11], 'cost': 114}, {'id': 23, 'elements': [7, 8], 'cost': 176}, {'id': 24, 'elements': [10, 11, 13, 14], 'cost': 212}, {'id': 25, 'elements': [1, 2, 4], 'cost': 261}, {'id': 26, 'elements': [12, 13, 14], 'cost': 258}, {'id': 27, 'elements': [7, 9, 10, 11], 'cost': 128}, {'id': 28, 'elements': [10, 12, 13], 'cost': 216}, {'id': 29, 'elements': [8, 9, 11, 12], 'cost': 256}, {'id': 30, 'elements': [11, 12, 13], 'cost': 123}, {'id': 31, 'elements': [5, 9], 'cost': 104}, {'id': 32, 'elements': [9, 10, 11, 13], 'cost': 176}, {'id': 33, 'elements': [12, 13], 'cost': 170}, {'id': 34, 'elements': [6, 9], 'cost': 58}, {'id': 35, 'elements': [7], 'cost': 38}, {'id': 36, 'elements': [4, 5, 7], 'cost': 54}, {'id': 37, 'elements': [10, 11, 12, 13], 'cost': 204}, {'id': 38, 'elements': [13, 14], 'cost': 28}, {'id': 39, 'elements': [10], 'cost': 56}, {'id': 40, 'elements': [7, 8, 9, 10], 'cost': 92}, {'id': 41, 'elements': [1, 3, 4], 'cost': 126}, {'id': 42, 'elements': [6, 7, 8], 'cost': 96}, {'id': 43, 'elements': [7, 9], 'cost': 132}, {'id': 44, 'elements': [11, 12], 'cost': 190}, {'id': 45, 'elements': [2, 4, 5, 6], 'cost': 356}, {'id': 46, 'elements': [1, 2, 4, 6], 'cost': 152}, {'id': 47, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [14], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0011_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0011_bag_stats.png'}","[5, 14, 17, 18, 34, 35]",427.0,"{'num_elements': 14, 'num_sets': 60, 'sets': [{'id': 'S1', 'elements': ['A', 'C'], 'cost': 80}, {'id': 'S2', 'elements': ['K', 'N'], 'cost': 118}, {'id': 'S3', 'elements': ['I', 'J', 'K'], 'cost': 54}, {'id': 'S4', 'elements': ['F', 'H'], 'cost': 104}, {'id': 'S5', 'elements': ['E'], 'cost': 17}, {'id': 'S6', 'elements': ['A', 'B'], 'cost': 94}, {'id': 'S7', 'elements': ['J', 'L', 'N'], 'cost': 207}, {'id': 'S8', 'elements': ['J', 'K', 'M'], 'cost': 9}, {'id': 'S9', 'elements': ['G', 'H', 'I'], 'cost': 6}, {'id': 'S10', 'elements': ['I', 'K'], 'cost': 160}, {'id': 'S11', 'elements': ['B', 'C', 'E'], 'cost': 240}, {'id': 'S12', 'elements': ['B', 'C'], 'cost': 156}, {'id': 'S13', 'elements': ['D', 'F'], 'cost': 128}, {'id': 'S14', 'elements': ['H', 'K', 'M', 'N'], 'cost': 60}, {'id': 'S15', 'elements': ['H', 'K', 'L', 'M'], 'cost': 372}, {'id': 'S16', 'elements': ['B', 'C', 'D', 'E'], 'cost': 152}, {'id': 'S17', 'elements': ['J', 'L'], 'cost': 26}, {'id': 'S18', 'elements': ['A', 'B', 'C', 'D'], 'cost': 228}, {'id': 'S19', 'elements': ['C', 'D', 'F', 'G'], 'cost': 208}, {'id': 'S20', 'elements': ['G', 'H', 'J'], 'cost': 162}, {'id': 'S21', 'elements': ['K', 'L', 'N'], 'cost': 63}, {'id': 'S22', 'elements': ['J', 'K'], 'cost': 114}, {'id': 'S23', 'elements': ['G', 'H'], 'cost': 176}, {'id': 'S24', 'elements': ['J', 'K', 'M', 'N'], 'cost': 212}, {'id': 'S25', 'elements': ['A', 'B', 'D'], 'cost': 261}, {'id': 'S26', 'elements': ['L', 'M', 'N'], 'cost': 258}, {'id': 'S27', 'elements': ['G', 'I', 'J', 'K'], 'cost': 128}, {'id': 'S28', 'elements': ['J', 'L', 'M'], 'cost': 216}, {'id': 'S29', 'elements': ['H', 'I', 'K', 'L'], 'cost': 256}, {'id': 'S30', 'elements': ['K', 'L', 'M'], 'cost': 123}, {'id': 'S31', 'elements': ['E', 'I'], 'cost': 104}, {'id': 'S32', 'elements': ['I', 'J', 'K', 'M'], 'cost': 176}, {'id': 'S33', 'elements': ['L', 'M'], 'cost': 170}, {'id': 'S34', 'elements': ['F', 'I'], 'cost': 58}, {'id': 'S35', 'elements': ['G'], 'cost': 38}, {'id': 'S36', 'elements': ['D', 'E', 'G'], 'cost': 54}, {'id': 'S37', 'elements': ['J', 'K', 'L', 'M'], 'cost': 204}, {'id': 'S38', 'elements': ['M', 'N'], 'cost': 28}, {'id': 'S39', 'elements': ['J'], 'cost': 56}, {'id': 'S40', 'elements': ['G', 'H', 'I', 'J'], 'cost': 92}, {'id': 'S41', 'elements': ['A', 'C', 'D'], 'cost': 126}, {'id': 'S42', 'elements': ['F', 'G', 'H'], 'cost': 96}, {'id': 'S43', 'elements': ['G', 'I'], 'cost': 132}, {'id': 'S44', 'elements': ['K', 'L'], 'cost': 190}, {'id': 'S45', 'elements': ['B', 'D', 'E', 'F'], 'cost': 356}, {'id': 'S46', 'elements': ['A', 'B', 'D', 'F'], 'cost': 152}, {'id': 'S47', 'elements': ['A'], 'cost': 10000}, {'id': 'S48', 'elements': ['B'], 'cost': 10000}, {'id': 'S49', 'elements': ['C'], 'cost': 10000}, {'id': 'S50', 'elements': ['D'], 'cost': 10000}, {'id': 'S51', 'elements': ['E'], 'cost': 10000}, {'id': 'S52', 'elements': ['F'], 'cost': 10000}, {'id': 'S53', 'elements': ['G'], 'cost': 10000}, {'id': 'S54', 'elements': ['H'], 'cost': 10000}, {'id': 'S55', 'elements': ['I'], 'cost': 10000}, {'id': 'S56', 'elements': ['J'], 'cost': 10000}, {'id': 'S57', 'elements': ['K'], 'cost': 10000}, {'id': 'S58', 'elements': ['L'], 'cost': 10000}, {'id': 'S59', 'elements': ['M'], 'cost': 10000}, {'id': 'S60', 'elements': ['N'], 'cost': 10000}]}","['S5', 'S14', 'S17', 'S18', 'S34', 'S35']",12,csv,names
SPP,SPP,"Recently the school looked at a bunch of possible study groups and asked which ones to run so every student ends up in exactly one chosen group; the clear aim is to keep the total coaching cost as low as possible — that total comes from summing the cost of each group that’s used. No student can be omitted or assigned to more than one group. The exact lists and costs are provided below.

{
  ""total_students"": 14,
  ""candidate_groups"": 65,
  ""sets"": [
    {
      ""group_id"": ""S1"",
      ""student_ids"": [
        12
      ],
      ""coaching_cost"": 79
    },
    {
      ""group_id"": ""S2"",
      ""student_ids"": [
        9,
        10,
        11,
        12
      ],
      ""coaching_cost"": 264
    },
    {
      ""group_id"": ""S3"",
      ""student_ids"": [
        9,
        11,
        12,
        13
      ],
      ""coaching_cost"": 216
    },
    {
      ""group_id"": ""S4"",
      ""student_ids"": [
        7,
        9,
        10,
        11
      ],
      ""coaching_cost"": 320
    },
    {
      ""group_id"": ""S5"",
      ""student_ids"": [
        10,
        11,
        12,
        13
      ],
      ""coaching_cost"": 332
    },
    {
      ""group_id"": ""S6"",
      ""student_ids"": [
        9,
        10
      ],
      ""coaching_cost"": 200
    },
    {
      ""group_id"": ""S7"",
      ""student_ids"": [
        2,
        3,
        4
      ],
      ""coaching_cost"": 72
    },
    {
      ""group_id"": ""S8"",
      ""student_ids"": [
        5,
        10
      ],
      ""coaching_cost"": 120
    },
    {
      ""group_id"": ""S9"",
      ""student_ids"": [
        0,
        2,
        4
      ],
      ""coaching_cost"": 300
    },
    {
      ""group_id"": ""S10"",
      ""student_ids"": [
        8,
        9,
        10,
        11
      ],
      ""coaching_cost"": 40
    },
    {
      ""group_id"": ""S11"",
      ""student_ids"": [
        4,
        5,
        7,
        8
      ],
      ""coaching_cost"": 84
    },
    {
      ""group_id"": ""S12"",
      ""student_ids"": [
        7,
        8,
        10
      ],
      ""coaching_cost"": 255
    },
    {
      ""group_id"": ""S13"",
      ""student_ids"": [
        4,
        7
      ],
      ""coaching_cost"": 160
    },
    {
      ""group_id"": ""S14"",
      ""student_ids"": [
        5,
        9,
        11
      ],
      ""coaching_cost"": 258
    },
    {
      ""group_id"": ""S15"",
      ""student_ids"": [
        3,
        5
      ],
      ""coaching_cost"": 160
    },
    {
      ""group_id"": ""S16"",
      ""student_ids"": [
        0,
        1
      ],
      ""coaching_cost"": 30
    },
    {
      ""group_id"": ""S17"",
      ""student_ids"": [
        5,
        9
      ],
      ""coaching_cost"": 134
    },
    {
      ""group_id"": ""S18"",
      ""student_ids"": [
        3,
        4,
        5,
        6
      ],
      ""coaching_cost"": 336
    },
    {
      ""group_id"": ""S19"",
      ""student_ids"": [
        6,
        7
      ],
      ""coaching_cost"": 198
    },
    {
      ""group_id"": ""S20"",
      ""student_ids"": [
        0,
        1,
        2
      ],
      ""coaching_cost"": 84
    },
    {
      ""group_id"": ""S21"",
      ""student_ids"": [
        5,
        6,
        7,
        8
      ],
      ""coaching_cost"": 380
    },
    {
      ""group_id"": ""S22"",
      ""student_ids"": [
        7,
        8,
        10,
        11
      ],
      ""coaching_cost"": 368
    },
    {
      ""group_id"": ""S23"",
      ""student_ids"": [
        5,
        6
      ],
      ""coaching_cost"": 82
    },
    {
      ""group_id"": ""S24"",
      ""student_ids"": [
        8
      ],
      ""coaching_cost"": 1
    },
    {
      ""group_id"": ""S25"",
      ""student_ids"": [
        6,
        7,
        10
      ],
      ""coaching_cost"": 30
    },
    {
      ""group_id"": ""S26"",
      ""student_ids"": [
        6,
        9
      ],
      ""coaching_cost"": 78
    },
    {
      ""group_id"": ""S27"",
      ""student_ids"": [
        3,
        4,
        5
      ],
      ""coaching_cost"": 255
    },
    {
      ""group_id"": ""S28"",
      ""student_ids"": [
        2
      ],
      ""coaching_cost"": 18
    },
    {
      ""group_id"": ""S29"",
      ""student_ids"": [
        11,
        12,
        13
      ],
      ""coaching_cost"": 57
    },
    {
      ""group_id"": ""S30"",
      ""student_ids"": [
        12,
        13
      ],
      ""coaching_cost"": 4
    },
    {
      ""group_id"": ""S31"",
      ""student_ids"": [
        5,
        7,
        9
      ],
      ""coaching_cost"": 153
    },
    {
      ""group_id"": ""S32"",
      ""student_ids"": [
        1,
        2,
        3
      ],
      ""coaching_cost"": 135
    },
    {
      ""group_id"": ""S33"",
      ""student_ids"": [
        8,
        11
      ],
      ""coaching_cost"": 48
    },
    {
      ""group_id"": ""S34"",
      ""student_ids"": [
        11,
        12
      ],
      ""coaching_cost"": 172
    },
    {
      ""group_id"": ""S35"",
      ""student_ids"": [
        6,
        8,
        9
      ],
      ""coaching_cost"": 93
    },
    {
      ""group_id"": ""S36"",
      ""student_ids"": [
        7,
        8
      ],
      ""coaching_cost"": 126
    },
    {
      ""group_id"": ""S37"",
      ""student_ids"": [
        7,
        9,
        10
      ],
      ""coaching_cost"": 219
    },
    {
      ""group_id"": ""S38"",
      ""student_ids"": [
        11,
        13
      ],
      ""coaching_cost"": 112
    },
    {
      ""group_id"": ""S39"",
      ""student_ids"": [
        3,
        6
      ],
      ""coaching_cost"": 124
    },
    {
      ""group_id"": ""S40"",
      ""student_ids"": [
        3,
        5,
        6
      ],
      ""coaching_cost"": 249
    },
    {
      ""group_id"": ""S41"",
      ""student_ids"": [
        9,
        12
      ],
      ""coaching_cost"": 20
    },
    {
      ""group_id"": ""S42"",
      ""student_ids"": [
        0,
        1,
        5
      ],
      ""coaching_cost"": 183
    },
    {
      ""group_id"": ""S43"",
      ""student_ids"": [
        1,
        2,
        4
      ],
      ""coaching_cost"": 282
    },
    {
      ""group_id"": ""S44"",
      ""student_ids"": [
        1,
        2,
        3,
        4
      ],
      ""coaching_cost"": 340
    },
    {
      ""group_id"": ""S45"",
      ""student_ids"": [
        1,
        2,
        3,
        5
      ],
      ""coaching_cost"": 76
    },
    {
      ""group_id"": ""S46"",
      ""student_ids"": [
        5,
        8,
        9
      ],
      ""coaching_cost"": 90
    },
    {
      ""group_id"": ""S47"",
      ""student_ids"": [
        4,
        6
      ],
      ""coaching_cost"": 134
    },
    {
      ""group_id"": ""S48"",
      ""student_ids"": [
        2,
        5,
        6,
        7
      ],
      ""coaching_cost"": 336
    },
    {
      ""group_id"": ""S49"",
      ""student_ids"": [
        10,
        11
      ],
      ""coaching_cost"": 46
    },
    {
      ""group_id"": ""S50"",
      ""student_ids"": [
        7,
        8,
        9,
        11
      ],
      ""coaching_cost"": 180
    },
    {
      ""group_id"": ""S51"",
      ""student_ids"": [
        4,
        8,
        9
      ],
      ""coaching_cost"": 171
    },
    {
      ""group_id"": ""S52"",
      ""student_ids"": [
        0
      ],
      ""coaching_cost"": 10000
    },
    {
      ""group_id"": ""S53"",
      ""student_ids"": [
        1
      ],
      ""coaching_cost"": 10000
    },
    {
      ""group_id"": ""S54"",
      ""student_ids"": [
        2
      ],
      ""coaching_cost"": 10000
    },
    {
      ""group_id"": ""S55"",
      ""student_ids"": [
        3
      ],
      ""coaching_cost"": 10000
    },
    {
      ""group_id"": ""S56"",
      ""student_ids"": [
        4
      ],
      ""coaching_cost"": 10000
    },
    {
      ""group_id"": ""S57"",
      ""student_ids"": [
        5
      ],
      ""coaching_cost"": 10000
    },
    {
      ""group_id"": ""S58"",
      ""student_ids"": [
        6
      ],
      ""coaching_cost"": 10000
    },
    {
      ""group_id"": ""S59"",
      ""student_ids"": [
        7
      ],
      ""coaching_cost"": 10000
    },
    {
      ""group_id"": ""S60"",
      ""student_ids"": [
        8
      ],
      ""coaching_cost"": 10000
    },
    {
      ""group_id"": ""S61"",
      ""student_ids"": [
        9
      ],
      ""coaching_cost"": 10000
    },
    {
      ""group_id"": ""S62"",
      ""student_ids"": [
        10
      ],
      ""coaching_cost"": 10000
    },
    {
      ""group_id"": ""S63"",
      ""student_ids"": [
        11
      ],
      ""coaching_cost"": 10000
    },
    {
      ""group_id"": ""S64"",
      ""student_ids"": [
        12
      ],
      ""coaching_cost"": 10000
    },
    {
      ""group_id"": ""S65"",
      ""student_ids"": [
        13
      ],
      ""coaching_cost"": 10000
    }
  ]
}

When you send the final choice, just paste it in that simple JSON shape so it's easy to read and check.

{
  ""solution"": [""group_id"", ...]
}

""solution"" is just the list of chosen study-group ids — one entry per selected group. Think of it like filling in a short form: replace the placeholder with the actual group ids from the instance. This JSON is only a sketch of the expected shape, not the real answer.

Please make sure to use the exact identifiers from the instance input — do not rename them or invent new labels.
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 14, 'num_sets': 65, 'density': 0.17142857142857143, 'sets': [{'id': 1, 'elements': [13], 'cost': 79}, {'id': 2, 'elements': [10, 11, 12, 13], 'cost': 264}, {'id': 3, 'elements': [10, 12, 13, 14], 'cost': 216}, {'id': 4, 'elements': [8, 10, 11, 12], 'cost': 320}, {'id': 5, 'elements': [11, 12, 13, 14], 'cost': 332}, {'id': 6, 'elements': [10, 11], 'cost': 200}, {'id': 7, 'elements': [3, 4, 5], 'cost': 72}, {'id': 8, 'elements': [6, 11], 'cost': 120}, {'id': 9, 'elements': [1, 3, 5], 'cost': 300}, {'id': 10, 'elements': [9, 10, 11, 12], 'cost': 40}, {'id': 11, 'elements': [5, 6, 8, 9], 'cost': 84}, {'id': 12, 'elements': [8, 9, 11], 'cost': 255}, {'id': 13, 'elements': [5, 8], 'cost': 160}, {'id': 14, 'elements': [6, 10, 12], 'cost': 258}, {'id': 15, 'elements': [4, 6], 'cost': 160}, {'id': 16, 'elements': [1, 2], 'cost': 30}, {'id': 17, 'elements': [6, 10], 'cost': 134}, {'id': 18, 'elements': [4, 5, 6, 7], 'cost': 336}, {'id': 19, 'elements': [7, 8], 'cost': 198}, {'id': 20, 'elements': [1, 2, 3], 'cost': 84}, {'id': 21, 'elements': [6, 7, 8, 9], 'cost': 380}, {'id': 22, 'elements': [8, 9, 11, 12], 'cost': 368}, {'id': 23, 'elements': [6, 7], 'cost': 82}, {'id': 24, 'elements': [9], 'cost': 1}, {'id': 25, 'elements': [7, 8, 11], 'cost': 30}, {'id': 26, 'elements': [7, 10], 'cost': 78}, {'id': 27, 'elements': [4, 5, 6], 'cost': 255}, {'id': 28, 'elements': [3], 'cost': 18}, {'id': 29, 'elements': [12, 13, 14], 'cost': 57}, {'id': 30, 'elements': [13, 14], 'cost': 4}, {'id': 31, 'elements': [6, 8, 10], 'cost': 153}, {'id': 32, 'elements': [2, 3, 4], 'cost': 135}, {'id': 33, 'elements': [9, 12], 'cost': 48}, {'id': 34, 'elements': [12, 13], 'cost': 172}, {'id': 35, 'elements': [7, 9, 10], 'cost': 93}, {'id': 36, 'elements': [8, 9], 'cost': 126}, {'id': 37, 'elements': [8, 10, 11], 'cost': 219}, {'id': 38, 'elements': [12, 14], 'cost': 112}, {'id': 39, 'elements': [4, 7], 'cost': 124}, {'id': 40, 'elements': [4, 6, 7], 'cost': 249}, {'id': 41, 'elements': [10, 13], 'cost': 20}, {'id': 42, 'elements': [1, 2, 6], 'cost': 183}, {'id': 43, 'elements': [2, 3, 5], 'cost': 282}, {'id': 44, 'elements': [2, 3, 4, 5], 'cost': 340}, {'id': 45, 'elements': [2, 3, 4, 6], 'cost': 76}, {'id': 46, 'elements': [6, 9, 10], 'cost': 90}, {'id': 47, 'elements': [5, 7], 'cost': 134}, {'id': 48, 'elements': [3, 6, 7, 8], 'cost': 336}, {'id': 49, 'elements': [11, 12], 'cost': 46}, {'id': 50, 'elements': [8, 9, 10, 12], 'cost': 180}, {'id': 51, 'elements': [5, 9, 10], 'cost': 171}, {'id': 52, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [14], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0012_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0012_bag_stats.png'}","[7, 16, 25, 29, 46]",279.0,"{'num_elements': 14, 'num_sets': 65, 'sets': [{'id': 'S1', 'elements': [12], 'cost': 79}, {'id': 'S2', 'elements': [9, 10, 11, 12], 'cost': 264}, {'id': 'S3', 'elements': [9, 11, 12, 13], 'cost': 216}, {'id': 'S4', 'elements': [7, 9, 10, 11], 'cost': 320}, {'id': 'S5', 'elements': [10, 11, 12, 13], 'cost': 332}, {'id': 'S6', 'elements': [9, 10], 'cost': 200}, {'id': 'S7', 'elements': [2, 3, 4], 'cost': 72}, {'id': 'S8', 'elements': [5, 10], 'cost': 120}, {'id': 'S9', 'elements': [0, 2, 4], 'cost': 300}, {'id': 'S10', 'elements': [8, 9, 10, 11], 'cost': 40}, {'id': 'S11', 'elements': [4, 5, 7, 8], 'cost': 84}, {'id': 'S12', 'elements': [7, 8, 10], 'cost': 255}, {'id': 'S13', 'elements': [4, 7], 'cost': 160}, {'id': 'S14', 'elements': [5, 9, 11], 'cost': 258}, {'id': 'S15', 'elements': [3, 5], 'cost': 160}, {'id': 'S16', 'elements': [0, 1], 'cost': 30}, {'id': 'S17', 'elements': [5, 9], 'cost': 134}, {'id': 'S18', 'elements': [3, 4, 5, 6], 'cost': 336}, {'id': 'S19', 'elements': [6, 7], 'cost': 198}, {'id': 'S20', 'elements': [0, 1, 2], 'cost': 84}, {'id': 'S21', 'elements': [5, 6, 7, 8], 'cost': 380}, {'id': 'S22', 'elements': [7, 8, 10, 11], 'cost': 368}, {'id': 'S23', 'elements': [5, 6], 'cost': 82}, {'id': 'S24', 'elements': [8], 'cost': 1}, {'id': 'S25', 'elements': [6, 7, 10], 'cost': 30}, {'id': 'S26', 'elements': [6, 9], 'cost': 78}, {'id': 'S27', 'elements': [3, 4, 5], 'cost': 255}, {'id': 'S28', 'elements': [2], 'cost': 18}, {'id': 'S29', 'elements': [11, 12, 13], 'cost': 57}, {'id': 'S30', 'elements': [12, 13], 'cost': 4}, {'id': 'S31', 'elements': [5, 7, 9], 'cost': 153}, {'id': 'S32', 'elements': [1, 2, 3], 'cost': 135}, {'id': 'S33', 'elements': [8, 11], 'cost': 48}, {'id': 'S34', 'elements': [11, 12], 'cost': 172}, {'id': 'S35', 'elements': [6, 8, 9], 'cost': 93}, {'id': 'S36', 'elements': [7, 8], 'cost': 126}, {'id': 'S37', 'elements': [7, 9, 10], 'cost': 219}, {'id': 'S38', 'elements': [11, 13], 'cost': 112}, {'id': 'S39', 'elements': [3, 6], 'cost': 124}, {'id': 'S40', 'elements': [3, 5, 6], 'cost': 249}, {'id': 'S41', 'elements': [9, 12], 'cost': 20}, {'id': 'S42', 'elements': [0, 1, 5], 'cost': 183}, {'id': 'S43', 'elements': [1, 2, 4], 'cost': 282}, {'id': 'S44', 'elements': [1, 2, 3, 4], 'cost': 340}, {'id': 'S45', 'elements': [1, 2, 3, 5], 'cost': 76}, {'id': 'S46', 'elements': [5, 8, 9], 'cost': 90}, {'id': 'S47', 'elements': [4, 6], 'cost': 134}, {'id': 'S48', 'elements': [2, 5, 6, 7], 'cost': 336}, {'id': 'S49', 'elements': [10, 11], 'cost': 46}, {'id': 'S50', 'elements': [7, 8, 9, 11], 'cost': 180}, {'id': 'S51', 'elements': [4, 8, 9], 'cost': 171}, {'id': 'S52', 'elements': [0], 'cost': 10000}, {'id': 'S53', 'elements': [1], 'cost': 10000}, {'id': 'S54', 'elements': [2], 'cost': 10000}, {'id': 'S55', 'elements': [3], 'cost': 10000}, {'id': 'S56', 'elements': [4], 'cost': 10000}, {'id': 'S57', 'elements': [5], 'cost': 10000}, {'id': 'S58', 'elements': [6], 'cost': 10000}, {'id': 'S59', 'elements': [7], 'cost': 10000}, {'id': 'S60', 'elements': [8], 'cost': 10000}, {'id': 'S61', 'elements': [9], 'cost': 10000}, {'id': 'S62', 'elements': [10], 'cost': 10000}, {'id': 'S63', 'elements': [11], 'cost': 10000}, {'id': 'S64', 'elements': [12], 'cost': 10000}, {'id': 'S65', 'elements': [13], 'cost': 10000}]}","['S7', 'S16', 'S25', 'S29', 'S46']",13,json,0
SPP,SPP,"At a small house renovation, the owner is choosing from a set of contractor deals where each deal takes care of some rooms and has a fee. The job is to assign exactly one deal to every room (so no room is left out or covered more than once) and then compute the total by adding the fees of the selected deals — the best assignment is the one with the lowest combined cost. The particular packages and their room coverage are given below.

There are 14 distinct rooms and 54 available packages listed below.
Package S1 covers rooms 12 13 and has a fee of 108.
Package S2 covers rooms 10 11 12 13 and has a fee of 332.
Package S3 covers rooms 1 2 3 4 and has a fee of 264.
Package S4 covers rooms 8 9 and has a fee of 48.
Package S5 covers rooms 4 7 8 and has a fee of 279.
Package S6 covers rooms 12 13 14 and has a fee of 183.
Package S7 covers rooms 8 9 10 11 and has a fee of 236.
Package S8 covers rooms 1 2 3 and has a fee of 78.
Package S9 covers rooms 9 10 11 and has a fee of 69.
Package S10 covers rooms 9 10 12 and has a fee of 291.
Package S11 covers rooms 9 10 and has a fee of 140.
Package S12 covers rooms 11 12 13 14 and has a fee of 360.
Package S13 covers rooms 4 5 6 and has a fee of 135.
Package S14 covers rooms 4 5 7 and has a fee of 111.
Package S15 covers rooms 7 9 10 and has a fee of 3.
Package S16 covers rooms 5 6 7 8 and has a fee of 276.
Package S17 covers rooms 5 6 9 and has a fee of 81.
Package S18 covers rooms 11 13 14 and has a fee of 186.
Package S19 covers rooms 5 6 and has a fee of 102.
Package S20 covers rooms 1 3 4 5 and has a fee of 316.
Package S21 covers rooms 10 12 13 and has a fee of 36.
Package S22 covers rooms 6 7 8 10 and has a fee of 44.
Package S23 covers rooms 11 12 14 and has a fee of 201.
Package S24 covers rooms 8 11 and has a fee of 46.
Package S25 covers rooms 5 9 10 and has a fee of 201.
Package S26 covers rooms 1 2 and has a fee of 180.
Package S27 covers rooms 2 4 and has a fee of 168.
Package S28 covers rooms 1 and has a fee of 36.
Package S29 covers rooms 3 4 5 7 and has a fee of 364.
Package S30 covers rooms 2 3 and has a fee of 144.
Package S31 covers rooms 4 5 7 9 and has a fee of 236.
Package S32 covers rooms 6 9 10 11 and has a fee of 260.
Package S33 covers rooms 3 5 and has a fee of 138.
Package S34 covers rooms 6 8 and has a fee of 118.
Package S35 covers rooms 9 10 11 12 and has a fee of 276.
Package S36 covers rooms 13 and has a fee of 37.
Package S37 covers rooms 10 12 and has a fee of 142.
Package S38 covers rooms 6 7 8 9 and has a fee of 384.
Package S39 covers rooms 3 4 6 and has a fee of 102.
Package S40 covers rooms 11 14 and has a fee of 174.
Package S41 covers rooms 1 and has a fee of 10000.
Package S42 covers rooms 2 and has a fee of 10000.
Package S43 covers rooms 3 and has a fee of 10000.
Package S44 covers rooms 4 and has a fee of 10000.
Package S45 covers rooms 5 and has a fee of 10000.
Package S46 covers rooms 6 and has a fee of 10000.
Package S47 covers rooms 7 and has a fee of 10000.
Package S48 covers rooms 8 and has a fee of 10000.
Package S49 covers rooms 9 and has a fee of 10000.
Package S50 covers rooms 10 and has a fee of 10000.
Package S51 covers rooms 11 and has a fee of 10000.
Package S52 covers rooms 12 and has a fee of 10000.
Package S53 covers rooms 13 and has a fee of 10000.
Package S54 covers rooms 14 and has a fee of 10000.
Assign exactly one package to each of the 14 rooms so the owner’s total fee (the sum of chosen package fees) is minimized.

If you want to return the chosen contractor deals, just send it in this simple JSON shape.

{
  ""solution"": [""deal_id"", ...]
}

This says: put the IDs of the selected deals in the solution list (one entry per chosen package). It's just a sketch of the shape I expect — not the actual answer itself.

Please make sure you use the exact identifiers from the problem instance, without renaming or inventing new labels. 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 14, 'num_sets': 54, 'density': 0.17195767195767195, 'sets': [{'id': 1, 'elements': [12, 13], 'cost': 108}, {'id': 2, 'elements': [10, 11, 12, 13], 'cost': 332}, {'id': 3, 'elements': [1, 2, 3, 4], 'cost': 264}, {'id': 4, 'elements': [8, 9], 'cost': 48}, {'id': 5, 'elements': [4, 7, 8], 'cost': 279}, {'id': 6, 'elements': [12, 13, 14], 'cost': 183}, {'id': 7, 'elements': [8, 9, 10, 11], 'cost': 236}, {'id': 8, 'elements': [1, 2, 3], 'cost': 78}, {'id': 9, 'elements': [9, 10, 11], 'cost': 69}, {'id': 10, 'elements': [9, 10, 12], 'cost': 291}, {'id': 11, 'elements': [9, 10], 'cost': 140}, {'id': 12, 'elements': [11, 12, 13, 14], 'cost': 360}, {'id': 13, 'elements': [4, 5, 6], 'cost': 135}, {'id': 14, 'elements': [4, 5, 7], 'cost': 111}, {'id': 15, 'elements': [7, 9, 10], 'cost': 3}, {'id': 16, 'elements': [5, 6, 7, 8], 'cost': 276}, {'id': 17, 'elements': [5, 6, 9], 'cost': 81}, {'id': 18, 'elements': [11, 13, 14], 'cost': 186}, {'id': 19, 'elements': [5, 6], 'cost': 102}, {'id': 20, 'elements': [1, 3, 4, 5], 'cost': 316}, {'id': 21, 'elements': [10, 12, 13], 'cost': 36}, {'id': 22, 'elements': [6, 7, 8, 10], 'cost': 44}, {'id': 23, 'elements': [11, 12, 14], 'cost': 201}, {'id': 24, 'elements': [8, 11], 'cost': 46}, {'id': 25, 'elements': [5, 9, 10], 'cost': 201}, {'id': 26, 'elements': [1, 2], 'cost': 180}, {'id': 27, 'elements': [2, 4], 'cost': 168}, {'id': 28, 'elements': [1], 'cost': 36}, {'id': 29, 'elements': [3, 4, 5, 7], 'cost': 364}, {'id': 30, 'elements': [2, 3], 'cost': 144}, {'id': 31, 'elements': [4, 5, 7, 9], 'cost': 236}, {'id': 32, 'elements': [6, 9, 10, 11], 'cost': 260}, {'id': 33, 'elements': [3, 5], 'cost': 138}, {'id': 34, 'elements': [6, 8], 'cost': 118}, {'id': 35, 'elements': [9, 10, 11, 12], 'cost': 276}, {'id': 36, 'elements': [13], 'cost': 37}, {'id': 37, 'elements': [10, 12], 'cost': 142}, {'id': 38, 'elements': [6, 7, 8, 9], 'cost': 384}, {'id': 39, 'elements': [3, 4, 6], 'cost': 102}, {'id': 40, 'elements': [11, 14], 'cost': 174}, {'id': 41, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 42, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 43, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 44, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 45, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 46, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 47, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [14], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0013_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0013_bag_stats.png'}","[6, 8, 13, 15, 24]",445.0,"{'num_elements': 14, 'num_sets': 54, 'sets': [{'id': 'S1', 'elements': [12, 13], 'cost': 108}, {'id': 'S2', 'elements': [10, 11, 12, 13], 'cost': 332}, {'id': 'S3', 'elements': [1, 2, 3, 4], 'cost': 264}, {'id': 'S4', 'elements': [8, 9], 'cost': 48}, {'id': 'S5', 'elements': [4, 7, 8], 'cost': 279}, {'id': 'S6', 'elements': [12, 13, 14], 'cost': 183}, {'id': 'S7', 'elements': [8, 9, 10, 11], 'cost': 236}, {'id': 'S8', 'elements': [1, 2, 3], 'cost': 78}, {'id': 'S9', 'elements': [9, 10, 11], 'cost': 69}, {'id': 'S10', 'elements': [9, 10, 12], 'cost': 291}, {'id': 'S11', 'elements': [9, 10], 'cost': 140}, {'id': 'S12', 'elements': [11, 12, 13, 14], 'cost': 360}, {'id': 'S13', 'elements': [4, 5, 6], 'cost': 135}, {'id': 'S14', 'elements': [4, 5, 7], 'cost': 111}, {'id': 'S15', 'elements': [7, 9, 10], 'cost': 3}, {'id': 'S16', 'elements': [5, 6, 7, 8], 'cost': 276}, {'id': 'S17', 'elements': [5, 6, 9], 'cost': 81}, {'id': 'S18', 'elements': [11, 13, 14], 'cost': 186}, {'id': 'S19', 'elements': [5, 6], 'cost': 102}, {'id': 'S20', 'elements': [1, 3, 4, 5], 'cost': 316}, {'id': 'S21', 'elements': [10, 12, 13], 'cost': 36}, {'id': 'S22', 'elements': [6, 7, 8, 10], 'cost': 44}, {'id': 'S23', 'elements': [11, 12, 14], 'cost': 201}, {'id': 'S24', 'elements': [8, 11], 'cost': 46}, {'id': 'S25', 'elements': [5, 9, 10], 'cost': 201}, {'id': 'S26', 'elements': [1, 2], 'cost': 180}, {'id': 'S27', 'elements': [2, 4], 'cost': 168}, {'id': 'S28', 'elements': [1], 'cost': 36}, {'id': 'S29', 'elements': [3, 4, 5, 7], 'cost': 364}, {'id': 'S30', 'elements': [2, 3], 'cost': 144}, {'id': 'S31', 'elements': [4, 5, 7, 9], 'cost': 236}, {'id': 'S32', 'elements': [6, 9, 10, 11], 'cost': 260}, {'id': 'S33', 'elements': [3, 5], 'cost': 138}, {'id': 'S34', 'elements': [6, 8], 'cost': 118}, {'id': 'S35', 'elements': [9, 10, 11, 12], 'cost': 276}, {'id': 'S36', 'elements': [13], 'cost': 37}, {'id': 'S37', 'elements': [10, 12], 'cost': 142}, {'id': 'S38', 'elements': [6, 7, 8, 9], 'cost': 384}, {'id': 'S39', 'elements': [3, 4, 6], 'cost': 102}, {'id': 'S40', 'elements': [11, 14], 'cost': 174}, {'id': 'S41', 'elements': [1], 'cost': 10000}, {'id': 'S42', 'elements': [2], 'cost': 10000}, {'id': 'S43', 'elements': [3], 'cost': 10000}, {'id': 'S44', 'elements': [4], 'cost': 10000}, {'id': 'S45', 'elements': [5], 'cost': 10000}, {'id': 'S46', 'elements': [6], 'cost': 10000}, {'id': 'S47', 'elements': [7], 'cost': 10000}, {'id': 'S48', 'elements': [8], 'cost': 10000}, {'id': 'S49', 'elements': [9], 'cost': 10000}, {'id': 'S50', 'elements': [10], 'cost': 10000}, {'id': 'S51', 'elements': [11], 'cost': 10000}, {'id': 'S52', 'elements': [12], 'cost': 10000}, {'id': 'S53', 'elements': [13], 'cost': 10000}, {'id': 'S54', 'elements': [14], 'cost': 10000}]}","['S6', 'S8', 'S13', 'S15', 'S24']",14,nl,1
SPP,SPP,"Many people don’t realize that choosing which bundles to publish is a bit like solving a puzzle: each bundle covers some shows and has a licensing fee. The task is to pick a set of bundles so every show is in exactly one published bundle — no show missing, no show appearing twice. The best pick is the one with the smallest total licensing bill, which you find by adding the fees of the bundles you chose. The detailed list of bundles and costs appears below.

- **num_shows_total**: 18
- **num_bundles_available**: 79

| bundle_id | licensing_fee | shows_in_bundle |
|---|---|---|
| S1 | 96 | 16 17 |
| S2 | 96 | 13 14 16 17 |
| S3 | 310 | 12 14 15 16 17 |
| S4 | 450 | 0 1 2 3 6 |
| S5 | 168 | 6 8 10 11 |
| S6 | 237 | 6 7 8 |
| S7 | 144 | 3 5 7 |
| S8 | 222 | 10 11 12 |
| S9 | 104 | 3 4 5 6 |
| S10 | 108 | 0 4 |
| S11 | 292 | 13 14 15 16 |
| S12 | 360 | 12 13 14 15 16 |
| S13 | 370 | 6 7 8 9 10 |
| S14 | 252 | 14 15 16 17 |
| S15 | 110 | 11 12 13 15 16 |
| S16 | 44 | 9 10 11 13 |
| S17 | 182 | 6 8 |
| S18 | 141 | 11 12 14 |
| S19 | 376 | 12 13 15 17 |
| S20 | 66 | 5 9 11 |
| S21 | 12 | 1 3 5 |
| S22 | 250 | 13 14 15 16 17 |
| S23 | 204 | 4 6 8 |
| S24 | 350 | 0 1 2 3 4 |
| S25 | 280 | 11 12 13 14 |
| S26 | 117 | 1 2 3 |
| S27 | 291 | 14 15 16 |
| S28 | 284 | 0 1 2 4 |
| S29 | 276 | 11 12 15 |
| S30 | 268 | 9 10 13 14 |
| S31 | 120 | 7 8 9 10 11 |
| S32 | 70 | 3 4 |
| S33 | 96 | 11 12 13 15 |
| S34 | 320 | 6 7 8 10 11 |
| S35 | 180 | 12 15 17 |
| S36 | 400 | 0 1 3 5 |
| S37 | 147 | 15 16 17 |
| S38 | 25 | 1 2 4 6 7 |
| S39 | 264 | 2 3 7 |
| S40 | 310 | 7 8 10 11 12 |
| S41 | 124 | 3 5 7 10 |
| S42 | 128 | 6 9 10 11 |
| S43 | 178 | 2 3 |
| S44 | 176 | 10 11 12 13 |
| S45 | 99 | 2 6 7 |
| S46 | 284 | 9 11 12 13 |
| S47 | 68 | 14 15 |
| S48 | 12 | 8 12 13 15 |
| S49 | 148 | 10 11 13 14 |
| S50 | 51 | 7 8 10 |
| S51 | 26 | 0 2 |
| S52 | 180 | 12 13 |
| S53 | 108 | 0 1 2 5 |
| S54 | 216 | 1 2 3 4 |
| S55 | 297 | 5 6 7 |
| S56 | 96 | 8 10 11 12 |
| S57 | 159 | 10 12 15 |
| S58 | 355 | 11 12 13 14 15 |
| S59 | 185 | 1 4 7 8 12 |
| S60 | 112 | 11 14 16 17 |
| S61 | 300 | 2 3 4 5 |
| S62 | 10000 | 0 |
| S63 | 10000 | 1 |
| S64 | 10000 | 2 |
| S65 | 10000 | 3 |
| S66 | 10000 | 4 |
| S67 | 10000 | 5 |
| S68 | 10000 | 6 |
| S69 | 10000 | 7 |
| S70 | 10000 | 8 |
| S71 | 10000 | 9 |
| S72 | 10000 | 10 |
| S73 | 10000 | 11 |
| S74 | 10000 | 12 |
| S75 | 10000 | 13 |
| S76 | 10000 | 14 |
| S77 | 10000 | 15 |
| S78 | 10000 | 16 |
| S79 | 10000 | 17 |

If you could return your final choice in a simple JSON form, that makes it super easy for me to read. Something like this:

{
  ""solution"": [""bundle_id"", ...]
}

This just means ""solution"" is a list of the bundle IDs you want to publish — each string in the array is the ID of one chosen bundle. It's a sketch of the shape I expect, not your actual answer; replace the placeholder entries with the real bundle identifiers from the instance.

Please be careful to use the identifiers exactly as they appear in the input — no renaming and no new labels.
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 18, 'num_sets': 79, 'density': 0.170182841068917, 'sets': [{'id': 1, 'elements': [17, 18], 'cost': 96}, {'id': 2, 'elements': [14, 15, 17, 18], 'cost': 96}, {'id': 3, 'elements': [13, 15, 16, 17, 18], 'cost': 310}, {'id': 4, 'elements': [1, 2, 3, 4, 7], 'cost': 450}, {'id': 5, 'elements': [7, 9, 11, 12], 'cost': 168}, {'id': 6, 'elements': [7, 8, 9], 'cost': 237}, {'id': 7, 'elements': [4, 6, 8], 'cost': 144}, {'id': 8, 'elements': [11, 12, 13], 'cost': 222}, {'id': 9, 'elements': [4, 5, 6, 7], 'cost': 104}, {'id': 10, 'elements': [1, 5], 'cost': 108}, {'id': 11, 'elements': [14, 15, 16, 17], 'cost': 292}, {'id': 12, 'elements': [13, 14, 15, 16, 17], 'cost': 360}, {'id': 13, 'elements': [7, 8, 9, 10, 11], 'cost': 370}, {'id': 14, 'elements': [15, 16, 17, 18], 'cost': 252}, {'id': 15, 'elements': [12, 13, 14, 16, 17], 'cost': 110}, {'id': 16, 'elements': [10, 11, 12, 14], 'cost': 44}, {'id': 17, 'elements': [7, 9], 'cost': 182}, {'id': 18, 'elements': [12, 13, 15], 'cost': 141}, {'id': 19, 'elements': [13, 14, 16, 18], 'cost': 376}, {'id': 20, 'elements': [6, 10, 12], 'cost': 66}, {'id': 21, 'elements': [2, 4, 6], 'cost': 12}, {'id': 22, 'elements': [14, 15, 16, 17, 18], 'cost': 250}, {'id': 23, 'elements': [5, 7, 9], 'cost': 204}, {'id': 24, 'elements': [1, 2, 3, 4, 5], 'cost': 350}, {'id': 25, 'elements': [12, 13, 14, 15], 'cost': 280}, {'id': 26, 'elements': [2, 3, 4], 'cost': 117}, {'id': 27, 'elements': [15, 16, 17], 'cost': 291}, {'id': 28, 'elements': [1, 2, 3, 5], 'cost': 284}, {'id': 29, 'elements': [12, 13, 16], 'cost': 276}, {'id': 30, 'elements': [10, 11, 14, 15], 'cost': 268}, {'id': 31, 'elements': [8, 9, 10, 11, 12], 'cost': 120}, {'id': 32, 'elements': [4, 5], 'cost': 70}, {'id': 33, 'elements': [12, 13, 14, 16], 'cost': 96}, {'id': 34, 'elements': [7, 8, 9, 11, 12], 'cost': 320}, {'id': 35, 'elements': [13, 16, 18], 'cost': 180}, {'id': 36, 'elements': [1, 2, 4, 6], 'cost': 400}, {'id': 37, 'elements': [16, 17, 18], 'cost': 147}, {'id': 38, 'elements': [2, 3, 5, 7, 8], 'cost': 25}, {'id': 39, 'elements': [3, 4, 8], 'cost': 264}, {'id': 40, 'elements': [8, 9, 11, 12, 13], 'cost': 310}, {'id': 41, 'elements': [4, 6, 8, 11], 'cost': 124}, {'id': 42, 'elements': [7, 10, 11, 12], 'cost': 128}, {'id': 43, 'elements': [3, 4], 'cost': 178}, {'id': 44, 'elements': [11, 12, 13, 14], 'cost': 176}, {'id': 45, 'elements': [3, 7, 8], 'cost': 99}, {'id': 46, 'elements': [10, 12, 13, 14], 'cost': 284}, {'id': 47, 'elements': [15, 16], 'cost': 68}, {'id': 48, 'elements': [9, 13, 14, 16], 'cost': 12}, {'id': 49, 'elements': [11, 12, 14, 15], 'cost': 148}, {'id': 50, 'elements': [8, 9, 11], 'cost': 51}, {'id': 51, 'elements': [1, 3], 'cost': 26}, {'id': 52, 'elements': [13, 14], 'cost': 180}, {'id': 53, 'elements': [1, 2, 3, 6], 'cost': 108}, {'id': 54, 'elements': [2, 3, 4, 5], 'cost': 216}, {'id': 55, 'elements': [6, 7, 8], 'cost': 297}, {'id': 56, 'elements': [9, 11, 12, 13], 'cost': 96}, {'id': 57, 'elements': [11, 13, 16], 'cost': 159}, {'id': 58, 'elements': [12, 13, 14, 15, 16], 'cost': 355}, {'id': 59, 'elements': [2, 5, 8, 9, 13], 'cost': 185}, {'id': 60, 'elements': [12, 15, 17, 18], 'cost': 112}, {'id': 61, 'elements': [3, 4, 5, 6], 'cost': 300}, {'id': 62, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 66, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 67, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 68, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 69, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 70, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 71, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 72, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 73, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 74, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 75, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 76, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 77, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 78, 'elements': [17], 'cost': 10000, 'singleton': True}, {'id': 79, 'elements': [18], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0014_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0014_bag_stats.png'}","[3, 6, 16, 32, 53]",769.0,"{'num_elements': 18, 'num_sets': 79, 'sets': [{'id': 'S1', 'elements': [16, 17], 'cost': 96}, {'id': 'S2', 'elements': [13, 14, 16, 17], 'cost': 96}, {'id': 'S3', 'elements': [12, 14, 15, 16, 17], 'cost': 310}, {'id': 'S4', 'elements': [0, 1, 2, 3, 6], 'cost': 450}, {'id': 'S5', 'elements': [6, 8, 10, 11], 'cost': 168}, {'id': 'S6', 'elements': [6, 7, 8], 'cost': 237}, {'id': 'S7', 'elements': [3, 5, 7], 'cost': 144}, {'id': 'S8', 'elements': [10, 11, 12], 'cost': 222}, {'id': 'S9', 'elements': [3, 4, 5, 6], 'cost': 104}, {'id': 'S10', 'elements': [0, 4], 'cost': 108}, {'id': 'S11', 'elements': [13, 14, 15, 16], 'cost': 292}, {'id': 'S12', 'elements': [12, 13, 14, 15, 16], 'cost': 360}, {'id': 'S13', 'elements': [6, 7, 8, 9, 10], 'cost': 370}, {'id': 'S14', 'elements': [14, 15, 16, 17], 'cost': 252}, {'id': 'S15', 'elements': [11, 12, 13, 15, 16], 'cost': 110}, {'id': 'S16', 'elements': [9, 10, 11, 13], 'cost': 44}, {'id': 'S17', 'elements': [6, 8], 'cost': 182}, {'id': 'S18', 'elements': [11, 12, 14], 'cost': 141}, {'id': 'S19', 'elements': [12, 13, 15, 17], 'cost': 376}, {'id': 'S20', 'elements': [5, 9, 11], 'cost': 66}, {'id': 'S21', 'elements': [1, 3, 5], 'cost': 12}, {'id': 'S22', 'elements': [13, 14, 15, 16, 17], 'cost': 250}, {'id': 'S23', 'elements': [4, 6, 8], 'cost': 204}, {'id': 'S24', 'elements': [0, 1, 2, 3, 4], 'cost': 350}, {'id': 'S25', 'elements': [11, 12, 13, 14], 'cost': 280}, {'id': 'S26', 'elements': [1, 2, 3], 'cost': 117}, {'id': 'S27', 'elements': [14, 15, 16], 'cost': 291}, {'id': 'S28', 'elements': [0, 1, 2, 4], 'cost': 284}, {'id': 'S29', 'elements': [11, 12, 15], 'cost': 276}, {'id': 'S30', 'elements': [9, 10, 13, 14], 'cost': 268}, {'id': 'S31', 'elements': [7, 8, 9, 10, 11], 'cost': 120}, {'id': 'S32', 'elements': [3, 4], 'cost': 70}, {'id': 'S33', 'elements': [11, 12, 13, 15], 'cost': 96}, {'id': 'S34', 'elements': [6, 7, 8, 10, 11], 'cost': 320}, {'id': 'S35', 'elements': [12, 15, 17], 'cost': 180}, {'id': 'S36', 'elements': [0, 1, 3, 5], 'cost': 400}, {'id': 'S37', 'elements': [15, 16, 17], 'cost': 147}, {'id': 'S38', 'elements': [1, 2, 4, 6, 7], 'cost': 25}, {'id': 'S39', 'elements': [2, 3, 7], 'cost': 264}, {'id': 'S40', 'elements': [7, 8, 10, 11, 12], 'cost': 310}, {'id': 'S41', 'elements': [3, 5, 7, 10], 'cost': 124}, {'id': 'S42', 'elements': [6, 9, 10, 11], 'cost': 128}, {'id': 'S43', 'elements': [2, 3], 'cost': 178}, {'id': 'S44', 'elements': [10, 11, 12, 13], 'cost': 176}, {'id': 'S45', 'elements': [2, 6, 7], 'cost': 99}, {'id': 'S46', 'elements': [9, 11, 12, 13], 'cost': 284}, {'id': 'S47', 'elements': [14, 15], 'cost': 68}, {'id': 'S48', 'elements': [8, 12, 13, 15], 'cost': 12}, {'id': 'S49', 'elements': [10, 11, 13, 14], 'cost': 148}, {'id': 'S50', 'elements': [7, 8, 10], 'cost': 51}, {'id': 'S51', 'elements': [0, 2], 'cost': 26}, {'id': 'S52', 'elements': [12, 13], 'cost': 180}, {'id': 'S53', 'elements': [0, 1, 2, 5], 'cost': 108}, {'id': 'S54', 'elements': [1, 2, 3, 4], 'cost': 216}, {'id': 'S55', 'elements': [5, 6, 7], 'cost': 297}, {'id': 'S56', 'elements': [8, 10, 11, 12], 'cost': 96}, {'id': 'S57', 'elements': [10, 12, 15], 'cost': 159}, {'id': 'S58', 'elements': [11, 12, 13, 14, 15], 'cost': 355}, {'id': 'S59', 'elements': [1, 4, 7, 8, 12], 'cost': 185}, {'id': 'S60', 'elements': [11, 14, 16, 17], 'cost': 112}, {'id': 'S61', 'elements': [2, 3, 4, 5], 'cost': 300}, {'id': 'S62', 'elements': [0], 'cost': 10000}, {'id': 'S63', 'elements': [1], 'cost': 10000}, {'id': 'S64', 'elements': [2], 'cost': 10000}, {'id': 'S65', 'elements': [3], 'cost': 10000}, {'id': 'S66', 'elements': [4], 'cost': 10000}, {'id': 'S67', 'elements': [5], 'cost': 10000}, {'id': 'S68', 'elements': [6], 'cost': 10000}, {'id': 'S69', 'elements': [7], 'cost': 10000}, {'id': 'S70', 'elements': [8], 'cost': 10000}, {'id': 'S71', 'elements': [9], 'cost': 10000}, {'id': 'S72', 'elements': [10], 'cost': 10000}, {'id': 'S73', 'elements': [11], 'cost': 10000}, {'id': 'S74', 'elements': [12], 'cost': 10000}, {'id': 'S75', 'elements': [13], 'cost': 10000}, {'id': 'S76', 'elements': [14], 'cost': 10000}, {'id': 'S77', 'elements': [15], 'cost': 10000}, {'id': 'S78', 'elements': [16], 'cost': 10000}, {'id': 'S79', 'elements': [17], 'cost': 10000}]}","['S3', 'S6', 'S16', 'S32', 'S53']",15,markdown_table,0
SPP,SPP,"At the prep center the team has to choose which box combinations will go on the menu; the constraint is that each ingredient must be contained in one and only one chosen box (nothing missing, nothing doubled), and the best lineup is the one with the lowest total procurement charge, computed by adding the costs of the picked boxes. The detailed options and their prices follow below.

# total_ingredients=14
# candidate_box_count=54
box_id,procurement_cost,box_ingredients
S1,168,0 1 2 3
S2,52,1 2 3 4
S3,300,2 4 5
S4,112,11 13
S5,234,11 12 13
S6,280,7 9 10 11
S7,36,6 7 9
S8,88,2 5
S9,68,9 10
S10,196,8 9
S11,148,7 8
S12,64,9 13
S13,148,12 13
S14,114,5 8
S15,144,5 6 8
S16,14,9 12
S17,74,10 11
S18,40,12
S19,210,0 1 2
S20,267,4 5 8
S21,99,0 1 3
S22,70,3 5
S23,170,8 12
S24,60,4 6
S25,276,6 8 10 11
S26,184,2 3
S27,40,5
S28,176,3 7
S29,16,3 4 5 6
S30,376,2 3 4 5
S31,54,8 10
S32,120,9 10 11
S33,297,8 9 10
S34,144,8 11
S35,282,9 11 13
S36,53,7
S37,144,4 5 6 7
S38,196,1 2
S39,352,7 8 9 10
S40,62,2
S41,10000,0
S42,10000,1
S43,10000,2
S44,10000,3
S45,10000,4
S46,10000,5
S47,10000,6
S48,10000,7
S49,10000,8
S50,10000,9
S51,10000,10
S52,10000,11
S53,10000,12
S54,10000,13

Also, when you send back the chosen lineup, please use this simple JSON shape so it's easy to parse:

{
  ""solution"": [""box_id"", ...]
}

This just says: put the ids of the boxes you picked into the solution list (one id per chosen box). Think of ""box_id"" as a placeholder for whatever box identifier shows up in the instance — the JSON above is just a sketch of the expected shape, not the actual answer.

A quick reminder: all identifiers must be used exactly as they appear in the instance input — no renaming and no new labels. 
- ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 14, 'num_sets': 54, 'density': 0.15343915343915343, 'sets': [{'id': 1, 'elements': [1, 2, 3, 4], 'cost': 168}, {'id': 2, 'elements': [2, 3, 4, 5], 'cost': 52}, {'id': 3, 'elements': [3, 5, 6], 'cost': 300}, {'id': 4, 'elements': [12, 14], 'cost': 112}, {'id': 5, 'elements': [12, 13, 14], 'cost': 234}, {'id': 6, 'elements': [8, 10, 11, 12], 'cost': 280}, {'id': 7, 'elements': [7, 8, 10], 'cost': 36}, {'id': 8, 'elements': [3, 6], 'cost': 88}, {'id': 9, 'elements': [10, 11], 'cost': 68}, {'id': 10, 'elements': [9, 10], 'cost': 196}, {'id': 11, 'elements': [8, 9], 'cost': 148}, {'id': 12, 'elements': [10, 14], 'cost': 64}, {'id': 13, 'elements': [13, 14], 'cost': 148}, {'id': 14, 'elements': [6, 9], 'cost': 114}, {'id': 15, 'elements': [6, 7, 9], 'cost': 144}, {'id': 16, 'elements': [10, 13], 'cost': 14}, {'id': 17, 'elements': [11, 12], 'cost': 74}, {'id': 18, 'elements': [13], 'cost': 40}, {'id': 19, 'elements': [1, 2, 3], 'cost': 210}, {'id': 20, 'elements': [5, 6, 9], 'cost': 267}, {'id': 21, 'elements': [1, 2, 4], 'cost': 99}, {'id': 22, 'elements': [4, 6], 'cost': 70}, {'id': 23, 'elements': [9, 13], 'cost': 170}, {'id': 24, 'elements': [5, 7], 'cost': 60}, {'id': 25, 'elements': [7, 9, 11, 12], 'cost': 276}, {'id': 26, 'elements': [3, 4], 'cost': 184}, {'id': 27, 'elements': [6], 'cost': 40}, {'id': 28, 'elements': [4, 8], 'cost': 176}, {'id': 29, 'elements': [4, 5, 6, 7], 'cost': 16}, {'id': 30, 'elements': [3, 4, 5, 6], 'cost': 376}, {'id': 31, 'elements': [9, 11], 'cost': 54}, {'id': 32, 'elements': [10, 11, 12], 'cost': 120}, {'id': 33, 'elements': [9, 10, 11], 'cost': 297}, {'id': 34, 'elements': [9, 12], 'cost': 144}, {'id': 35, 'elements': [10, 12, 14], 'cost': 282}, {'id': 36, 'elements': [8], 'cost': 53}, {'id': 37, 'elements': [5, 6, 7, 8], 'cost': 144}, {'id': 38, 'elements': [2, 3], 'cost': 196}, {'id': 39, 'elements': [8, 9, 10, 11], 'cost': 352}, {'id': 40, 'elements': [3], 'cost': 62}, {'id': 41, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 42, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 43, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 44, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 45, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 46, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 47, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [14], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0015_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0015_bag_stats.png'}","[4, 16, 19, 29, 31, 36]",459.0,"{'num_elements': 14, 'num_sets': 54, 'sets': [{'id': 'S1', 'elements': [0, 1, 2, 3], 'cost': 168}, {'id': 'S2', 'elements': [1, 2, 3, 4], 'cost': 52}, {'id': 'S3', 'elements': [2, 4, 5], 'cost': 300}, {'id': 'S4', 'elements': [11, 13], 'cost': 112}, {'id': 'S5', 'elements': [11, 12, 13], 'cost': 234}, {'id': 'S6', 'elements': [7, 9, 10, 11], 'cost': 280}, {'id': 'S7', 'elements': [6, 7, 9], 'cost': 36}, {'id': 'S8', 'elements': [2, 5], 'cost': 88}, {'id': 'S9', 'elements': [9, 10], 'cost': 68}, {'id': 'S10', 'elements': [8, 9], 'cost': 196}, {'id': 'S11', 'elements': [7, 8], 'cost': 148}, {'id': 'S12', 'elements': [9, 13], 'cost': 64}, {'id': 'S13', 'elements': [12, 13], 'cost': 148}, {'id': 'S14', 'elements': [5, 8], 'cost': 114}, {'id': 'S15', 'elements': [5, 6, 8], 'cost': 144}, {'id': 'S16', 'elements': [9, 12], 'cost': 14}, {'id': 'S17', 'elements': [10, 11], 'cost': 74}, {'id': 'S18', 'elements': [12], 'cost': 40}, {'id': 'S19', 'elements': [0, 1, 2], 'cost': 210}, {'id': 'S20', 'elements': [4, 5, 8], 'cost': 267}, {'id': 'S21', 'elements': [0, 1, 3], 'cost': 99}, {'id': 'S22', 'elements': [3, 5], 'cost': 70}, {'id': 'S23', 'elements': [8, 12], 'cost': 170}, {'id': 'S24', 'elements': [4, 6], 'cost': 60}, {'id': 'S25', 'elements': [6, 8, 10, 11], 'cost': 276}, {'id': 'S26', 'elements': [2, 3], 'cost': 184}, {'id': 'S27', 'elements': [5], 'cost': 40}, {'id': 'S28', 'elements': [3, 7], 'cost': 176}, {'id': 'S29', 'elements': [3, 4, 5, 6], 'cost': 16}, {'id': 'S30', 'elements': [2, 3, 4, 5], 'cost': 376}, {'id': 'S31', 'elements': [8, 10], 'cost': 54}, {'id': 'S32', 'elements': [9, 10, 11], 'cost': 120}, {'id': 'S33', 'elements': [8, 9, 10], 'cost': 297}, {'id': 'S34', 'elements': [8, 11], 'cost': 144}, {'id': 'S35', 'elements': [9, 11, 13], 'cost': 282}, {'id': 'S36', 'elements': [7], 'cost': 53}, {'id': 'S37', 'elements': [4, 5, 6, 7], 'cost': 144}, {'id': 'S38', 'elements': [1, 2], 'cost': 196}, {'id': 'S39', 'elements': [7, 8, 9, 10], 'cost': 352}, {'id': 'S40', 'elements': [2], 'cost': 62}, {'id': 'S41', 'elements': [0], 'cost': 10000}, {'id': 'S42', 'elements': [1], 'cost': 10000}, {'id': 'S43', 'elements': [2], 'cost': 10000}, {'id': 'S44', 'elements': [3], 'cost': 10000}, {'id': 'S45', 'elements': [4], 'cost': 10000}, {'id': 'S46', 'elements': [5], 'cost': 10000}, {'id': 'S47', 'elements': [6], 'cost': 10000}, {'id': 'S48', 'elements': [7], 'cost': 10000}, {'id': 'S49', 'elements': [8], 'cost': 10000}, {'id': 'S50', 'elements': [9], 'cost': 10000}, {'id': 'S51', 'elements': [10], 'cost': 10000}, {'id': 'S52', 'elements': [11], 'cost': 10000}, {'id': 'S53', 'elements': [12], 'cost': 10000}, {'id': 'S54', 'elements': [13], 'cost': 10000}]}","['S4', 'S16', 'S19', 'S29', 'S31', 'S36']",16,csv,0
SPP,SPP,"There’s a volunteer coordinator juggling multiple tasks and dozens of candidate teams. The coordinator needs to decide which teams to sign on so that every task is covered by exactly one chosen team — no task missing, no task handled by two teams — and each chosen team comes with a coordination cost. The quality of any choice is judged by adding up those team costs, and the preferable choices are the ones with the lowest total bill. The concrete details are shown below.

{
  ""num_tasks"": 14,
  ""num_candidate_teams"": 50,
  ""sets"": [
    {
      ""team_id"": ""S1"",
      ""tasks_covered"": [
        ""N""
      ],
      ""coordination_cost"": 35
    },
    {
      ""team_id"": ""S2"",
      ""tasks_covered"": [
        ""B"",
        ""D"",
        ""E"",
        ""G""
      ],
      ""coordination_cost"": 228
    },
    {
      ""team_id"": ""S3"",
      ""tasks_covered"": [
        ""G"",
        ""H"",
        ""I"",
        ""J""
      ],
      ""coordination_cost"": 400
    },
    {
      ""team_id"": ""S4"",
      ""tasks_covered"": [
        ""L"",
        ""M""
      ],
      ""coordination_cost"": 102
    },
    {
      ""team_id"": ""S5"",
      ""tasks_covered"": [
        ""K"",
        ""L"",
        ""M""
      ],
      ""coordination_cost"": 273
    },
    {
      ""team_id"": ""S6"",
      ""tasks_covered"": [
        ""G"",
        ""I"",
        ""J"",
        ""K""
      ],
      ""coordination_cost"": 128
    },
    {
      ""team_id"": ""S7"",
      ""tasks_covered"": [
        ""E"",
        ""G"",
        ""J""
      ],
      ""coordination_cost"": 294
    },
    {
      ""team_id"": ""S8"",
      ""tasks_covered"": [
        ""A"",
        ""B"",
        ""D""
      ],
      ""coordination_cost"": 300
    },
    {
      ""team_id"": ""S9"",
      ""tasks_covered"": [
        ""E"",
        ""G"",
        ""H"",
        ""I""
      ],
      ""coordination_cost"": 400
    },
    {
      ""team_id"": ""S10"",
      ""tasks_covered"": [
        ""H""
      ],
      ""coordination_cost"": 92
    },
    {
      ""team_id"": ""S11"",
      ""tasks_covered"": [
        ""F"",
        ""H"",
        ""I"",
        ""J""
      ],
      ""coordination_cost"": 84
    },
    {
      ""team_id"": ""S12"",
      ""tasks_covered"": [
        ""A"",
        ""B"",
        ""C""
      ],
      ""coordination_cost"": 33
    },
    {
      ""team_id"": ""S13"",
      ""tasks_covered"": [
        ""G"",
        ""H"",
        ""J""
      ],
      ""coordination_cost"": 63
    },
    {
      ""team_id"": ""S14"",
      ""tasks_covered"": [
        ""E"",
        ""F""
      ],
      ""coordination_cost"": 86
    },
    {
      ""team_id"": ""S15"",
      ""tasks_covered"": [
        ""I"",
        ""J"",
        ""L""
      ],
      ""coordination_cost"": 297
    },
    {
      ""team_id"": ""S16"",
      ""tasks_covered"": [
        ""F"",
        ""H"",
        ""J""
      ],
      ""coordination_cost"": 111
    },
    {
      ""team_id"": ""S17"",
      ""tasks_covered"": [
        ""A"",
        ""D"",
        ""E""
      ],
      ""coordination_cost"": 294
    },
    {
      ""team_id"": ""S18"",
      ""tasks_covered"": [
        ""G"",
        ""H""
      ],
      ""coordination_cost"": 2
    },
    {
      ""team_id"": ""S19"",
      ""tasks_covered"": [
        ""G"",
        ""H"",
        ""I""
      ],
      ""coordination_cost"": 159
    },
    {
      ""team_id"": ""S20"",
      ""tasks_covered"": [
        ""I"",
        ""J"",
        ""K""
      ],
      ""coordination_cost"": 240
    },
    {
      ""team_id"": ""S21"",
      ""tasks_covered"": [
        ""K"",
        ""L"",
        ""N""
      ],
      ""coordination_cost"": 252
    },
    {
      ""team_id"": ""S22"",
      ""tasks_covered"": [
        ""C"",
        ""D""
      ],
      ""coordination_cost"": 156
    },
    {
      ""team_id"": ""S23"",
      ""tasks_covered"": [
        ""A"",
        ""B""
      ],
      ""coordination_cost"": 46
    },
    {
      ""team_id"": ""S24"",
      ""tasks_covered"": [
        ""L"",
        ""M"",
        ""N""
      ],
      ""coordination_cost"": 141
    },
    {
      ""team_id"": ""S25"",
      ""tasks_covered"": [
        ""A"",
        ""B"",
        ""C"",
        ""D""
      ],
      ""coordination_cost"": 392
    },
    {
      ""team_id"": ""S26"",
      ""tasks_covered"": [
        ""E"",
        ""F"",
        ""G"",
        ""H""
      ],
      ""coordination_cost"": 56
    },
    {
      ""team_id"": ""S27"",
      ""tasks_covered"": [
        ""I"",
        ""K"",
        ""L""
      ],
      ""coordination_cost"": 234
    },
    {
      ""team_id"": ""S28"",
      ""tasks_covered"": [
        ""J"",
        ""K"",
        ""L"",
        ""M""
      ],
      ""coordination_cost"": 320
    },
    {
      ""team_id"": ""S29"",
      ""tasks_covered"": [
        ""F"",
        ""H"",
        ""I""
      ],
      ""coordination_cost"": 273
    },
    {
      ""team_id"": ""S30"",
      ""tasks_covered"": [
        ""E""
      ],
      ""coordination_cost"": 33
    },
    {
      ""team_id"": ""S31"",
      ""tasks_covered"": [
        ""C"",
        ""D"",
        ""F"",
        ""G""
      ],
      ""coordination_cost"": 172
    },
    {
      ""team_id"": ""S32"",
      ""tasks_covered"": [
        ""L"",
        ""N""
      ],
      ""coordination_cost"": 46
    },
    {
      ""team_id"": ""S33"",
      ""tasks_covered"": [
        ""M"",
        ""N""
      ],
      ""coordination_cost"": 122
    },
    {
      ""team_id"": ""S34"",
      ""tasks_covered"": [
        ""I"",
        ""J"",
        ""K"",
        ""M""
      ],
      ""coordination_cost"": 172
    },
    {
      ""team_id"": ""S35"",
      ""tasks_covered"": [
        ""K""
      ],
      ""coordination_cost"": 46
    },
    {
      ""team_id"": ""S36"",
      ""tasks_covered"": [
        ""I"",
        ""J"",
        ""K"",
        ""L""
      ],
      ""coordination_cost"": 60
    },
    {
      ""team_id"": ""S37"",
      ""tasks_covered"": [
        ""A""
      ],
      ""coordination_cost"": 10000
    },
    {
      ""team_id"": ""S38"",
      ""tasks_covered"": [
        ""B""
      ],
      ""coordination_cost"": 10000
    },
    {
      ""team_id"": ""S39"",
      ""tasks_covered"": [
        ""C""
      ],
      ""coordination_cost"": 10000
    },
    {
      ""team_id"": ""S40"",
      ""tasks_covered"": [
        ""D""
      ],
      ""coordination_cost"": 10000
    },
    {
      ""team_id"": ""S41"",
      ""tasks_covered"": [
        ""E""
      ],
      ""coordination_cost"": 10000
    },
    {
      ""team_id"": ""S42"",
      ""tasks_covered"": [
        ""F""
      ],
      ""coordination_cost"": 10000
    },
    {
      ""team_id"": ""S43"",
      ""tasks_covered"": [
        ""G""
      ],
      ""coordination_cost"": 10000
    },
    {
      ""team_id"": ""S44"",
      ""tasks_covered"": [
        ""H""
      ],
      ""coordination_cost"": 10000
    },
    {
      ""team_id"": ""S45"",
      ""tasks_covered"": [
        ""I""
      ],
      ""coordination_cost"": 10000
    },
    {
      ""team_id"": ""S46"",
      ""tasks_covered"": [
        ""J""
      ],
      ""coordination_cost"": 10000
    },
    {
      ""team_id"": ""S47"",
      ""tasks_covered"": [
        ""K""
      ],
      ""coordination_cost"": 10000
    },
    {
      ""team_id"": ""S48"",
      ""tasks_covered"": [
        ""L""
      ],
      ""coordination_cost"": 10000
    },
    {
      ""team_id"": ""S49"",
      ""tasks_covered"": [
        ""M""
      ],
      ""coordination_cost"": 10000
    },
    {
      ""team_id"": ""S50"",
      ""tasks_covered"": [
        ""N""
      ],
      ""coordination_cost"": 10000
    }
  ]
}

If you want my pick, please send it back in this simple shape — nothing fancy, just a tiny JSON list of the chosen teams:

{
  ""solution"": [""team_id"", ...]
}

This just means: put the IDs of the teams you want in the solution array (each entry is a team identifier). Think of it like ticking boxes on a form — the array holds the team IDs you’re signing on. The JSON above is just a sketch of the shape I expect, not the actual answer.

Please make sure you use the exact identifiers from the instance input — don’t rename them or invent new ones. Valid identifiers look like plain numbers such as ""1"" or ""23"", single capital letters like ""A"" or ""B"", or a capital letter followed by digits like ""A1"" or ""X7"".","{'problem_type': 'SPP', 'num_elements': 14, 'num_sets': 50, 'density': 0.16857142857142857, 'sets': [{'id': 1, 'elements': [14], 'cost': 35}, {'id': 2, 'elements': [2, 4, 5, 7], 'cost': 228}, {'id': 3, 'elements': [7, 8, 9, 10], 'cost': 400}, {'id': 4, 'elements': [12, 13], 'cost': 102}, {'id': 5, 'elements': [11, 12, 13], 'cost': 273}, {'id': 6, 'elements': [7, 9, 10, 11], 'cost': 128}, {'id': 7, 'elements': [5, 7, 10], 'cost': 294}, {'id': 8, 'elements': [1, 2, 4], 'cost': 300}, {'id': 9, 'elements': [5, 7, 8, 9], 'cost': 400}, {'id': 10, 'elements': [8], 'cost': 92}, {'id': 11, 'elements': [6, 8, 9, 10], 'cost': 84}, {'id': 12, 'elements': [1, 2, 3], 'cost': 33}, {'id': 13, 'elements': [7, 8, 10], 'cost': 63}, {'id': 14, 'elements': [5, 6], 'cost': 86}, {'id': 15, 'elements': [9, 10, 12], 'cost': 297}, {'id': 16, 'elements': [6, 8, 10], 'cost': 111}, {'id': 17, 'elements': [1, 4, 5], 'cost': 294}, {'id': 18, 'elements': [7, 8], 'cost': 2}, {'id': 19, 'elements': [7, 8, 9], 'cost': 159}, {'id': 20, 'elements': [9, 10, 11], 'cost': 240}, {'id': 21, 'elements': [11, 12, 14], 'cost': 252}, {'id': 22, 'elements': [3, 4], 'cost': 156}, {'id': 23, 'elements': [1, 2], 'cost': 46}, {'id': 24, 'elements': [12, 13, 14], 'cost': 141}, {'id': 25, 'elements': [1, 2, 3, 4], 'cost': 392}, {'id': 26, 'elements': [5, 6, 7, 8], 'cost': 56}, {'id': 27, 'elements': [9, 11, 12], 'cost': 234}, {'id': 28, 'elements': [10, 11, 12, 13], 'cost': 320}, {'id': 29, 'elements': [6, 8, 9], 'cost': 273}, {'id': 30, 'elements': [5], 'cost': 33}, {'id': 31, 'elements': [3, 4, 6, 7], 'cost': 172}, {'id': 32, 'elements': [12, 14], 'cost': 46}, {'id': 33, 'elements': [13, 14], 'cost': 122}, {'id': 34, 'elements': [9, 10, 11, 13], 'cost': 172}, {'id': 35, 'elements': [11], 'cost': 46}, {'id': 36, 'elements': [9, 10, 11, 12], 'cost': 60}, {'id': 37, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 38, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 39, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 40, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 41, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 42, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 43, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 44, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 45, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 46, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 47, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [14], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0016_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0016_bag_stats.png'}","[22, 23, 26, 33, 36]",440.0,"{'num_elements': 14, 'num_sets': 50, 'sets': [{'id': 'S1', 'elements': ['N'], 'cost': 35}, {'id': 'S2', 'elements': ['B', 'D', 'E', 'G'], 'cost': 228}, {'id': 'S3', 'elements': ['G', 'H', 'I', 'J'], 'cost': 400}, {'id': 'S4', 'elements': ['L', 'M'], 'cost': 102}, {'id': 'S5', 'elements': ['K', 'L', 'M'], 'cost': 273}, {'id': 'S6', 'elements': ['G', 'I', 'J', 'K'], 'cost': 128}, {'id': 'S7', 'elements': ['E', 'G', 'J'], 'cost': 294}, {'id': 'S8', 'elements': ['A', 'B', 'D'], 'cost': 300}, {'id': 'S9', 'elements': ['E', 'G', 'H', 'I'], 'cost': 400}, {'id': 'S10', 'elements': ['H'], 'cost': 92}, {'id': 'S11', 'elements': ['F', 'H', 'I', 'J'], 'cost': 84}, {'id': 'S12', 'elements': ['A', 'B', 'C'], 'cost': 33}, {'id': 'S13', 'elements': ['G', 'H', 'J'], 'cost': 63}, {'id': 'S14', 'elements': ['E', 'F'], 'cost': 86}, {'id': 'S15', 'elements': ['I', 'J', 'L'], 'cost': 297}, {'id': 'S16', 'elements': ['F', 'H', 'J'], 'cost': 111}, {'id': 'S17', 'elements': ['A', 'D', 'E'], 'cost': 294}, {'id': 'S18', 'elements': ['G', 'H'], 'cost': 2}, {'id': 'S19', 'elements': ['G', 'H', 'I'], 'cost': 159}, {'id': 'S20', 'elements': ['I', 'J', 'K'], 'cost': 240}, {'id': 'S21', 'elements': ['K', 'L', 'N'], 'cost': 252}, {'id': 'S22', 'elements': ['C', 'D'], 'cost': 156}, {'id': 'S23', 'elements': ['A', 'B'], 'cost': 46}, {'id': 'S24', 'elements': ['L', 'M', 'N'], 'cost': 141}, {'id': 'S25', 'elements': ['A', 'B', 'C', 'D'], 'cost': 392}, {'id': 'S26', 'elements': ['E', 'F', 'G', 'H'], 'cost': 56}, {'id': 'S27', 'elements': ['I', 'K', 'L'], 'cost': 234}, {'id': 'S28', 'elements': ['J', 'K', 'L', 'M'], 'cost': 320}, {'id': 'S29', 'elements': ['F', 'H', 'I'], 'cost': 273}, {'id': 'S30', 'elements': ['E'], 'cost': 33}, {'id': 'S31', 'elements': ['C', 'D', 'F', 'G'], 'cost': 172}, {'id': 'S32', 'elements': ['L', 'N'], 'cost': 46}, {'id': 'S33', 'elements': ['M', 'N'], 'cost': 122}, {'id': 'S34', 'elements': ['I', 'J', 'K', 'M'], 'cost': 172}, {'id': 'S35', 'elements': ['K'], 'cost': 46}, {'id': 'S36', 'elements': ['I', 'J', 'K', 'L'], 'cost': 60}, {'id': 'S37', 'elements': ['A'], 'cost': 10000}, {'id': 'S38', 'elements': ['B'], 'cost': 10000}, {'id': 'S39', 'elements': ['C'], 'cost': 10000}, {'id': 'S40', 'elements': ['D'], 'cost': 10000}, {'id': 'S41', 'elements': ['E'], 'cost': 10000}, {'id': 'S42', 'elements': ['F'], 'cost': 10000}, {'id': 'S43', 'elements': ['G'], 'cost': 10000}, {'id': 'S44', 'elements': ['H'], 'cost': 10000}, {'id': 'S45', 'elements': ['I'], 'cost': 10000}, {'id': 'S46', 'elements': ['J'], 'cost': 10000}, {'id': 'S47', 'elements': ['K'], 'cost': 10000}, {'id': 'S48', 'elements': ['L'], 'cost': 10000}, {'id': 'S49', 'elements': ['M'], 'cost': 10000}, {'id': 'S50', 'elements': ['N'], 'cost': 10000}]}","['S22', 'S23', 'S26', 'S33', 'S36']",17,json,names
SPP,SPP,"There’s a pile of ad placement bundles on the table and a grid of ad slots that all need attention. Each bundle covers some of the slots and has a sticker price, so the task is to pick bundles that together cover every slot exactly once, and to keep the total bill down by summing the prices of the bundles chosen. The specific bundles, which slots they cover, and their costs are shown below.

There are 15 ad slots to fill and 66 bundles on the table:
Bundle S1 covers slots 1 2 and has a sticker price of 62.
Bundle S2 covers slots 12 13 15 and has a sticker price of 45.
Bundle S3 covers slots 1 2 3 and has a sticker price of 12.
Bundle S4 covers slots 12 13 14 15 and has a sticker price of 272.
Bundle S5 covers slots 4 8 and has a sticker price of 134.
Bundle S6 covers slots 1 2 3 5 and has a sticker price of 60.
Bundle S7 covers slots 12 15 and has a sticker price of 24.
Bundle S8 covers slots 12 14 and has a sticker price of 170.
Bundle S9 covers slots 1 2 3 4 and has a sticker price of 100.
Bundle S10 covers slots 8 9 and has a sticker price of 32.
Bundle S11 covers slots 7 9 10 and has a sticker price of 105.
Bundle S12 covers slots 12 13 and has a sticker price of 44.
Bundle S13 covers slots 10 12 and has a sticker price of 14.
Bundle S14 covers slots 6 10 and has a sticker price of 100.
Bundle S15 covers slots 7 8 and has a sticker price of 152.
Bundle S16 covers slots 11 12 13 and has a sticker price of 78.
Bundle S17 covers slots 9 11 12 13 and has a sticker price of 292.
Bundle S18 covers slots 14 15 and has a sticker price of 26.
Bundle S19 covers slots 10 11 12 and has a sticker price of 87.
Bundle S20 covers slots 11 13 14 and has a sticker price of 126.
Bundle S21 covers slots 2 3 5 6 and has a sticker price of 368.
Bundle S22 covers slots 4 5 6 7 and has a sticker price of 304.
Bundle S23 covers slots 8 9 11 and has a sticker price of 27.
Bundle S24 covers slots 13 14 15 and has a sticker price of 219.
Bundle S25 covers slots 7 9 10 11 and has a sticker price of 368.
Bundle S26 covers slots 6 7 9 and has a sticker price of 39.
Bundle S27 covers slots 2 3 4 5 and has a sticker price of 308.
Bundle S28 covers slots 4 5 6 8 and has a sticker price of 72.
Bundle S29 covers slots 3 5 6 and has a sticker price of 264.
Bundle S30 covers slots 4 5 7 8 and has a sticker price of 224.
Bundle S31 covers slots 11 12 14 15 and has a sticker price of 208.
Bundle S32 covers slots 10 13 14 15 and has a sticker price of 364.
Bundle S33 covers slots 3 4 5 and has a sticker price of 168.
Bundle S34 covers slots 5 6 7 8 and has a sticker price of 112.
Bundle S35 covers slots 6 8 10 and has a sticker price of 141.
Bundle S36 covers slots 2 3 4 and has a sticker price of 54.
Bundle S37 covers slots 10 11 12 13 and has a sticker price of 284.
Bundle S38 covers slots 3 4 5 7 and has a sticker price of 20.
Bundle S39 covers slots 9 10 11 13 and has a sticker price of 44.
Bundle S40 covers slots 6 7 8 10 and has a sticker price of 184.
Bundle S41 covers slots 11 12 13 14 and has a sticker price of 156.
Bundle S42 covers slots 9 10 11 12 and has a sticker price of 340.
Bundle S43 covers slots 10 11 13 14 and has a sticker price of 40.
Bundle S44 covers slots 3 4 5 6 and has a sticker price of 380.
Bundle S45 covers slots 4 5 and has a sticker price of 170.
Bundle S46 covers slots 11 12 14 and has a sticker price of 72.
Bundle S47 covers slots 11 13 14 15 and has a sticker price of 80.
Bundle S48 covers slots 1 4 6 and has a sticker price of 102.
Bundle S49 covers slots 13 14 and has a sticker price of 116.
Bundle S50 covers slots 3 7 9 and has a sticker price of 207.
Bundle S51 covers slots 4 5 6 and has a sticker price of 180.
Bundle S52 covers slots 1 and has a sticker price of 10000.
Bundle S53 covers slots 2 and has a sticker price of 10000.
Bundle S54 covers slots 3 and has a sticker price of 10000.
Bundle S55 covers slots 4 and has a sticker price of 10000.
Bundle S56 covers slots 5 and has a sticker price of 10000.
Bundle S57 covers slots 6 and has a sticker price of 10000.
Bundle S58 covers slots 7 and has a sticker price of 10000.
Bundle S59 covers slots 8 and has a sticker price of 10000.
Bundle S60 covers slots 9 and has a sticker price of 10000.
Bundle S61 covers slots 10 and has a sticker price of 10000.
Bundle S62 covers slots 11 and has a sticker price of 10000.
Bundle S63 covers slots 12 and has a sticker price of 10000.
Bundle S64 covers slots 13 and has a sticker price of 10000.
Bundle S65 covers slots 14 and has a sticker price of 10000.
Bundle S66 covers slots 15 and has a sticker price of 10000.
The objective is to cover every slot exactly once while minimizing the total sticker price.

If you want to reply with the chosen bundles, just follow this little JSON shape so it's easy to read and parse:

{
  ""solution"": [""bundle_id"", ...]
}

""solution"" is the list of bundle IDs you pick (each entry is the ID of one ad bundle that will cover some slots). The ""bundle_id"" text is a placeholder showing the kind of values you should put there — replace each placeholder with the actual bundle identifier from the instance when you answer. This JSON is just the sketch of the shape I want you to use, not the final selection itself.

Please make sure to use the exact identifiers from the instance input — don't rename them or invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'problem_type': 'SPP', 'num_elements': 15, 'num_sets': 66, 'density': 0.1797979797979798, 'sets': [{'id': 1, 'elements': [1, 2], 'cost': 62}, {'id': 2, 'elements': [12, 13, 15], 'cost': 45}, {'id': 3, 'elements': [1, 2, 3], 'cost': 12}, {'id': 4, 'elements': [12, 13, 14, 15], 'cost': 272}, {'id': 5, 'elements': [4, 8], 'cost': 134}, {'id': 6, 'elements': [1, 2, 3, 5], 'cost': 60}, {'id': 7, 'elements': [12, 15], 'cost': 24}, {'id': 8, 'elements': [12, 14], 'cost': 170}, {'id': 9, 'elements': [1, 2, 3, 4], 'cost': 100}, {'id': 10, 'elements': [8, 9], 'cost': 32}, {'id': 11, 'elements': [7, 9, 10], 'cost': 105}, {'id': 12, 'elements': [12, 13], 'cost': 44}, {'id': 13, 'elements': [10, 12], 'cost': 14}, {'id': 14, 'elements': [6, 10], 'cost': 100}, {'id': 15, 'elements': [7, 8], 'cost': 152}, {'id': 16, 'elements': [11, 12, 13], 'cost': 78}, {'id': 17, 'elements': [9, 11, 12, 13], 'cost': 292}, {'id': 18, 'elements': [14, 15], 'cost': 26}, {'id': 19, 'elements': [10, 11, 12], 'cost': 87}, {'id': 20, 'elements': [11, 13, 14], 'cost': 126}, {'id': 21, 'elements': [2, 3, 5, 6], 'cost': 368}, {'id': 22, 'elements': [4, 5, 6, 7], 'cost': 304}, {'id': 23, 'elements': [8, 9, 11], 'cost': 27}, {'id': 24, 'elements': [13, 14, 15], 'cost': 219}, {'id': 25, 'elements': [7, 9, 10, 11], 'cost': 368}, {'id': 26, 'elements': [6, 7, 9], 'cost': 39}, {'id': 27, 'elements': [2, 3, 4, 5], 'cost': 308}, {'id': 28, 'elements': [4, 5, 6, 8], 'cost': 72}, {'id': 29, 'elements': [3, 5, 6], 'cost': 264}, {'id': 30, 'elements': [4, 5, 7, 8], 'cost': 224}, {'id': 31, 'elements': [11, 12, 14, 15], 'cost': 208}, {'id': 32, 'elements': [10, 13, 14, 15], 'cost': 364}, {'id': 33, 'elements': [3, 4, 5], 'cost': 168}, {'id': 34, 'elements': [5, 6, 7, 8], 'cost': 112}, {'id': 35, 'elements': [6, 8, 10], 'cost': 141}, {'id': 36, 'elements': [2, 3, 4], 'cost': 54}, {'id': 37, 'elements': [10, 11, 12, 13], 'cost': 284}, {'id': 38, 'elements': [3, 4, 5, 7], 'cost': 20}, {'id': 39, 'elements': [9, 10, 11, 13], 'cost': 44}, {'id': 40, 'elements': [6, 7, 8, 10], 'cost': 184}, {'id': 41, 'elements': [11, 12, 13, 14], 'cost': 156}, {'id': 42, 'elements': [9, 10, 11, 12], 'cost': 340}, {'id': 43, 'elements': [10, 11, 13, 14], 'cost': 40}, {'id': 44, 'elements': [3, 4, 5, 6], 'cost': 380}, {'id': 45, 'elements': [4, 5], 'cost': 170}, {'id': 46, 'elements': [11, 12, 14], 'cost': 72}, {'id': 47, 'elements': [11, 13, 14, 15], 'cost': 80}, {'id': 48, 'elements': [1, 4, 6], 'cost': 102}, {'id': 49, 'elements': [13, 14], 'cost': 116}, {'id': 50, 'elements': [3, 7, 9], 'cost': 207}, {'id': 51, 'elements': [4, 5, 6], 'cost': 180}, {'id': 52, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 66, 'elements': [15], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0017_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0017_bag_stats.png'}","[1, 12, 14, 18, 23, 38]",279.0,"{'num_elements': 15, 'num_sets': 66, 'sets': [{'id': 'S1', 'elements': [1, 2], 'cost': 62}, {'id': 'S2', 'elements': [12, 13, 15], 'cost': 45}, {'id': 'S3', 'elements': [1, 2, 3], 'cost': 12}, {'id': 'S4', 'elements': [12, 13, 14, 15], 'cost': 272}, {'id': 'S5', 'elements': [4, 8], 'cost': 134}, {'id': 'S6', 'elements': [1, 2, 3, 5], 'cost': 60}, {'id': 'S7', 'elements': [12, 15], 'cost': 24}, {'id': 'S8', 'elements': [12, 14], 'cost': 170}, {'id': 'S9', 'elements': [1, 2, 3, 4], 'cost': 100}, {'id': 'S10', 'elements': [8, 9], 'cost': 32}, {'id': 'S11', 'elements': [7, 9, 10], 'cost': 105}, {'id': 'S12', 'elements': [12, 13], 'cost': 44}, {'id': 'S13', 'elements': [10, 12], 'cost': 14}, {'id': 'S14', 'elements': [6, 10], 'cost': 100}, {'id': 'S15', 'elements': [7, 8], 'cost': 152}, {'id': 'S16', 'elements': [11, 12, 13], 'cost': 78}, {'id': 'S17', 'elements': [9, 11, 12, 13], 'cost': 292}, {'id': 'S18', 'elements': [14, 15], 'cost': 26}, {'id': 'S19', 'elements': [10, 11, 12], 'cost': 87}, {'id': 'S20', 'elements': [11, 13, 14], 'cost': 126}, {'id': 'S21', 'elements': [2, 3, 5, 6], 'cost': 368}, {'id': 'S22', 'elements': [4, 5, 6, 7], 'cost': 304}, {'id': 'S23', 'elements': [8, 9, 11], 'cost': 27}, {'id': 'S24', 'elements': [13, 14, 15], 'cost': 219}, {'id': 'S25', 'elements': [7, 9, 10, 11], 'cost': 368}, {'id': 'S26', 'elements': [6, 7, 9], 'cost': 39}, {'id': 'S27', 'elements': [2, 3, 4, 5], 'cost': 308}, {'id': 'S28', 'elements': [4, 5, 6, 8], 'cost': 72}, {'id': 'S29', 'elements': [3, 5, 6], 'cost': 264}, {'id': 'S30', 'elements': [4, 5, 7, 8], 'cost': 224}, {'id': 'S31', 'elements': [11, 12, 14, 15], 'cost': 208}, {'id': 'S32', 'elements': [10, 13, 14, 15], 'cost': 364}, {'id': 'S33', 'elements': [3, 4, 5], 'cost': 168}, {'id': 'S34', 'elements': [5, 6, 7, 8], 'cost': 112}, {'id': 'S35', 'elements': [6, 8, 10], 'cost': 141}, {'id': 'S36', 'elements': [2, 3, 4], 'cost': 54}, {'id': 'S37', 'elements': [10, 11, 12, 13], 'cost': 284}, {'id': 'S38', 'elements': [3, 4, 5, 7], 'cost': 20}, {'id': 'S39', 'elements': [9, 10, 11, 13], 'cost': 44}, {'id': 'S40', 'elements': [6, 7, 8, 10], 'cost': 184}, {'id': 'S41', 'elements': [11, 12, 13, 14], 'cost': 156}, {'id': 'S42', 'elements': [9, 10, 11, 12], 'cost': 340}, {'id': 'S43', 'elements': [10, 11, 13, 14], 'cost': 40}, {'id': 'S44', 'elements': [3, 4, 5, 6], 'cost': 380}, {'id': 'S45', 'elements': [4, 5], 'cost': 170}, {'id': 'S46', 'elements': [11, 12, 14], 'cost': 72}, {'id': 'S47', 'elements': [11, 13, 14, 15], 'cost': 80}, {'id': 'S48', 'elements': [1, 4, 6], 'cost': 102}, {'id': 'S49', 'elements': [13, 14], 'cost': 116}, {'id': 'S50', 'elements': [3, 7, 9], 'cost': 207}, {'id': 'S51', 'elements': [4, 5, 6], 'cost': 180}, {'id': 'S52', 'elements': [1], 'cost': 10000}, {'id': 'S53', 'elements': [2], 'cost': 10000}, {'id': 'S54', 'elements': [3], 'cost': 10000}, {'id': 'S55', 'elements': [4], 'cost': 10000}, {'id': 'S56', 'elements': [5], 'cost': 10000}, {'id': 'S57', 'elements': [6], 'cost': 10000}, {'id': 'S58', 'elements': [7], 'cost': 10000}, {'id': 'S59', 'elements': [8], 'cost': 10000}, {'id': 'S60', 'elements': [9], 'cost': 10000}, {'id': 'S61', 'elements': [10], 'cost': 10000}, {'id': 'S62', 'elements': [11], 'cost': 10000}, {'id': 'S63', 'elements': [12], 'cost': 10000}, {'id': 'S64', 'elements': [13], 'cost': 10000}, {'id': 'S65', 'elements': [14], 'cost': 10000}, {'id': 'S66', 'elements': [15], 'cost': 10000}]}","['S1', 'S12', 'S14', 'S18', 'S23', 'S38']",18,nl,1
SPP,SPP,"There’s a production puzzle on the table: a stack of components that all need to be produced and a bunch of candidate batch runs that each make some of those components. The job is to pick a selection of those runs so every component is produced exactly once — no leftovers, no duplicates. Each run carries a cost, so once a selection is made the total expense is simply all those run costs added together, and the aim is to keep that total as small as possible. The specific parts, available batches, and their prices appear below.

{
  ""total_components"": 18,
  ""total_candidate_batches"": 91,
  ""sets"": [
    {
      ""batch_id"": ""S1"",
      ""produced_components"": [
        4,
        6,
        7,
        11
      ],
      ""batch_cost"": 8
    },
    {
      ""batch_id"": ""S2"",
      ""produced_components"": [
        0,
        1,
        2,
        3
      ],
      ""batch_cost"": 116
    },
    {
      ""batch_id"": ""S3"",
      ""produced_components"": [
        3,
        6,
        7,
        9
      ],
      ""batch_cost"": 312
    },
    {
      ""batch_id"": ""S4"",
      ""produced_components"": [
        6,
        7
      ],
      ""batch_cost"": 66
    },
    {
      ""batch_id"": ""S5"",
      ""produced_components"": [
        1,
        3,
        5
      ],
      ""batch_cost"": 165
    },
    {
      ""batch_id"": ""S6"",
      ""produced_components"": [
        7,
        8,
        9,
        12
      ],
      ""batch_cost"": 200
    },
    {
      ""batch_id"": ""S7"",
      ""produced_components"": [
        0,
        1,
        3
      ],
      ""batch_cost"": 27
    },
    {
      ""batch_id"": ""S8"",
      ""produced_components"": [
        3,
        4,
        6,
        7
      ],
      ""batch_cost"": 396
    },
    {
      ""batch_id"": ""S9"",
      ""produced_components"": [
        1,
        4,
        5
      ],
      ""batch_cost"": 30
    },
    {
      ""batch_id"": ""S10"",
      ""produced_components"": [
        4,
        9
      ],
      ""batch_cost"": 128
    },
    {
      ""batch_id"": ""S11"",
      ""produced_components"": [
        1,
        3
      ],
      ""batch_cost"": 70
    },
    {
      ""batch_id"": ""S12"",
      ""produced_components"": [
        10,
        12
      ],
      ""batch_cost"": 132
    },
    {
      ""batch_id"": ""S13"",
      ""produced_components"": [
        14,
        15,
        16,
        17
      ],
      ""batch_cost"": 36
    },
    {
      ""batch_id"": ""S14"",
      ""produced_components"": [
        8,
        10,
        11,
        16
      ],
      ""batch_cost"": 332
    },
    {
      ""batch_id"": ""S15"",
      ""produced_components"": [
        15,
        17
      ],
      ""batch_cost"": 118
    },
    {
      ""batch_id"": ""S16"",
      ""produced_components"": [
        6,
        7,
        8,
        9,
        10
      ],
      ""batch_cost"": 450
    },
    {
      ""batch_id"": ""S17"",
      ""produced_components"": [
        0,
        2,
        4
      ],
      ""batch_cost"": 243
    },
    {
      ""batch_id"": ""S18"",
      ""produced_components"": [
        4,
        5,
        8
      ],
      ""batch_cost"": 228
    },
    {
      ""batch_id"": ""S19"",
      ""produced_components"": [
        9,
        11,
        12
      ],
      ""batch_cost"": 207
    },
    {
      ""batch_id"": ""S20"",
      ""produced_components"": [
        10,
        13,
        14,
        15,
        17
      ],
      ""batch_cost"": 260
    },
    {
      ""batch_id"": ""S21"",
      ""produced_components"": [
        15,
        16
      ],
      ""batch_cost"": 138
    },
    {
      ""batch_id"": ""S22"",
      ""produced_components"": [
        4,
        10
      ],
      ""batch_cost"": 38
    },
    {
      ""batch_id"": ""S23"",
      ""produced_components"": [
        6,
        8
      ],
      ""batch_cost"": 26
    },
    {
      ""batch_id"": ""S24"",
      ""produced_components"": [
        2,
        6,
        7
      ],
      ""batch_cost"": 45
    },
    {
      ""batch_id"": ""S25"",
      ""produced_components"": [
        4,
        6,
        7,
        9
      ],
      ""batch_cost"": 380
    },
    {
      ""batch_id"": ""S26"",
      ""produced_components"": [
        8,
        9,
        10,
        11
      ],
      ""batch_cost"": 112
    },
    {
      ""batch_id"": ""S27"",
      ""produced_components"": [
        3,
        4,
        5,
        6,
        7
      ],
      ""batch_cost"": 355
    },
    {
      ""batch_id"": ""S28"",
      ""produced_components"": [
        8,
        13
      ],
      ""batch_cost"": 126
    },
    {
      ""batch_id"": ""S29"",
      ""produced_components"": [
        0,
        1
      ],
      ""batch_cost"": 168
    },
    {
      ""batch_id"": ""S30"",
      ""produced_components"": [
        7,
        8,
        11
      ],
      ""batch_cost"": 237
    },
    {
      ""batch_id"": ""S31"",
      ""produced_components"": [
        2,
        5,
        6,
        7
      ],
      ""batch_cost"": 156
    },
    {
      ""batch_id"": ""S32"",
      ""produced_components"": [
        0,
        1,
        5
      ],
      ""batch_cost"": 39
    },
    {
      ""batch_id"": ""S33"",
      ""produced_components"": [
        13,
        15,
        16,
        17
      ],
      ""batch_cost"": 28
    },
    {
      ""batch_id"": ""S34"",
      ""produced_components"": [
        5,
        7
      ],
      ""batch_cost"": 122
    },
    {
      ""batch_id"": ""S35"",
      ""produced_components"": [
        2,
        4,
        5,
        6
      ],
      ""batch_cost"": 224
    },
    {
      ""batch_id"": ""S36"",
      ""produced_components"": [
        11,
        15,
        17
      ],
      ""batch_cost"": 69
    },
    {
      ""batch_id"": ""S37"",
      ""produced_components"": [
        12,
        13,
        14,
        15,
        16
      ],
      ""batch_cost"": 500
    },
    {
      ""batch_id"": ""S38"",
      ""produced_components"": [
        5,
        8
      ],
      ""batch_cost"": 52
    },
    {
      ""batch_id"": ""S39"",
      ""produced_components"": [
        4,
        5,
        7
      ],
      ""batch_cost"": 168
    },
    {
      ""batch_id"": ""S40"",
      ""produced_components"": [
        12,
        14
      ],
      ""batch_cost"": 118
    },
    {
      ""batch_id"": ""S41"",
      ""produced_components"": [
        9,
        10,
        12,
        14
      ],
      ""batch_cost"": 32
    },
    {
      ""batch_id"": ""S42"",
      ""produced_components"": [
        9,
        10,
        11
      ],
      ""batch_cost"": 237
    },
    {
      ""batch_id"": ""S43"",
      ""produced_components"": [
        13,
        14,
        15,
        16,
        17
      ],
      ""batch_cost"": 380
    },
    {
      ""batch_id"": ""S44"",
      ""produced_components"": [
        12,
        13,
        15,
        16
      ],
      ""batch_cost"": 168
    },
    {
      ""batch_id"": ""S45"",
      ""produced_components"": [
        5,
        7,
        9
      ],
      ""batch_cost"": 222
    },
    {
      ""batch_id"": ""S46"",
      ""produced_components"": [
        9,
        10,
        13,
        14,
        16
      ],
      ""batch_cost"": 340
    },
    {
      ""batch_id"": ""S47"",
      ""produced_components"": [
        2,
        3,
        4
      ],
      ""batch_cost"": 6
    },
    {
      ""batch_id"": ""S48"",
      ""produced_components"": [
        11,
        12,
        14,
        16
      ],
      ""batch_cost"": 48
    },
    {
      ""batch_id"": ""S49"",
      ""produced_components"": [
        7,
        9
      ],
      ""batch_cost"": 148
    },
    {
      ""batch_id"": ""S50"",
      ""produced_components"": [
        0,
        1,
        3,
        4
      ],
      ""batch_cost"": 96
    },
    {
      ""batch_id"": ""S51"",
      ""produced_components"": [
        5,
        9,
        10,
        11,
        12
      ],
      ""batch_cost"": 445
    },
    {
      ""batch_id"": ""S52"",
      ""produced_components"": [
        1,
        2,
        3
      ],
      ""batch_cost"": 237
    },
    {
      ""batch_id"": ""S53"",
      ""produced_components"": [
        3,
        6,
        7,
        8
      ],
      ""batch_cost"": 32
    },
    {
      ""batch_id"": ""S54"",
      ""produced_components"": [
        9,
        11,
        12,
        13
      ],
      ""batch_cost"": 188
    },
    {
      ""batch_id"": ""S55"",
      ""produced_components"": [
        15,
        16,
        17
      ],
      ""batch_cost"": 54
    },
    {
      ""batch_id"": ""S56"",
      ""produced_components"": [
        5,
        6,
        7,
        9
      ],
      ""batch_cost"": 28
    },
    {
      ""batch_id"": ""S57"",
      ""produced_components"": [
        11,
        12
      ],
      ""batch_cost"": 32
    },
    {
      ""batch_id"": ""S58"",
      ""produced_components"": [
        14,
        16,
        17
      ],
      ""batch_cost"": 240
    },
    {
      ""batch_id"": ""S59"",
      ""produced_components"": [
        0,
        1,
        4
      ],
      ""batch_cost"": 153
    },
    {
      ""batch_id"": ""S60"",
      ""produced_components"": [
        6,
        7,
        8,
        9,
        12
      ],
      ""batch_cost"": 100
    },
    {
      ""batch_id"": ""S61"",
      ""produced_components"": [
        12,
        14,
        15,
        16,
        17
      ],
      ""batch_cost"": 305
    },
    {
      ""batch_id"": ""S62"",
      ""produced_components"": [
        10,
        12,
        15
      ],
      ""batch_cost"": 9
    },
    {
      ""batch_id"": ""S63"",
      ""produced_components"": [
        5,
        6,
        7,
        8,
        9
      ],
      ""batch_cost"": 210
    },
    {
      ""batch_id"": ""S64"",
      ""produced_components"": [
        13,
        14,
        15,
        17
      ],
      ""batch_cost"": 160
    },
    {
      ""batch_id"": ""S65"",
      ""produced_components"": [
        7,
        8
      ],
      ""batch_cost"": 74
    },
    {
      ""batch_id"": ""S66"",
      ""produced_components"": [
        0,
        1,
        2,
        3,
        5
      ],
      ""batch_cost"": 345
    },
    {
      ""batch_id"": ""S67"",
      ""produced_components"": [
        5,
        6,
        7
      ],
      ""batch_cost"": 60
    },
    {
      ""batch_id"": ""S68"",
      ""produced_components"": [
        0,
        2
      ],
      ""batch_cost"": 34
    },
    {
      ""batch_id"": ""S69"",
      ""produced_components"": [
        7,
        9,
        12
      ],
      ""batch_cost"": 123
    },
    {
      ""batch_id"": ""S70"",
      ""produced_components"": [
        2,
        3,
        4,
        5,
        8
      ],
      ""batch_cost"": 260
    },
    {
      ""batch_id"": ""S71"",
      ""produced_components"": [
        2,
        4,
        6,
        7,
        8
      ],
      ""batch_cost"": 325
    },
    {
      ""batch_id"": ""S72"",
      ""produced_components"": [
        8,
        9,
        12
      ],
      ""batch_cost"": 222
    },
    {
      ""batch_id"": ""S73"",
      ""produced_components"": [
        10,
        12,
        13,
        15
      ],
      ""batch_cost"": 348
    },
    {
      ""batch_id"": ""S74"",
      ""produced_components"": [
        0
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S75"",
      ""produced_components"": [
        1
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S76"",
      ""produced_components"": [
        2
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S77"",
      ""produced_components"": [
        3
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S78"",
      ""produced_components"": [
        4
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S79"",
      ""produced_components"": [
        5
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S80"",
      ""produced_components"": [
        6
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S81"",
      ""produced_components"": [
        7
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S82"",
      ""produced_components"": [
        8
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S83"",
      ""produced_components"": [
        9
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S84"",
      ""produced_components"": [
        10
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S85"",
      ""produced_components"": [
        11
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S86"",
      ""produced_components"": [
        12
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S87"",
      ""produced_components"": [
        13
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S88"",
      ""produced_components"": [
        14
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S89"",
      ""produced_components"": [
        15
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S90"",
      ""produced_components"": [
        16
      ],
      ""batch_cost"": 10000
    },
    {
      ""batch_id"": ""S91"",
      ""produced_components"": [
        17
      ],
      ""batch_cost"": 10000
    }
  ]
}

Oh, and when you give the final pick, please stick to this little JSON layout so it's easy to read and check:

{
  ""solution"": [""batch_id"", ...]
}

This is just a sketch of the shape I expect: ""solution"" holds a list of the batch/run identifiers you picked (one per produced component). Think of ""batch_id"" as a placeholder for whatever run label appears in the instance — it's just showing where each chosen run goes. The JSON is only the shape, not the actual answer.

Quick heads-up: use the identifiers exactly as they appear in the instance input — no renaming, no made-up labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'problem_type': 'SPP', 'num_elements': 18, 'num_sets': 91, 'density': 0.163003663003663, 'sets': [{'id': 1, 'elements': [5, 7, 8, 12], 'cost': 8}, {'id': 2, 'elements': [1, 2, 3, 4], 'cost': 116}, {'id': 3, 'elements': [4, 7, 8, 10], 'cost': 312}, {'id': 4, 'elements': [7, 8], 'cost': 66}, {'id': 5, 'elements': [2, 4, 6], 'cost': 165}, {'id': 6, 'elements': [8, 9, 10, 13], 'cost': 200}, {'id': 7, 'elements': [1, 2, 4], 'cost': 27}, {'id': 8, 'elements': [4, 5, 7, 8], 'cost': 396}, {'id': 9, 'elements': [2, 5, 6], 'cost': 30}, {'id': 10, 'elements': [5, 10], 'cost': 128}, {'id': 11, 'elements': [2, 4], 'cost': 70}, {'id': 12, 'elements': [11, 13], 'cost': 132}, {'id': 13, 'elements': [15, 16, 17, 18], 'cost': 36}, {'id': 14, 'elements': [9, 11, 12, 17], 'cost': 332}, {'id': 15, 'elements': [16, 18], 'cost': 118}, {'id': 16, 'elements': [7, 8, 9, 10, 11], 'cost': 450}, {'id': 17, 'elements': [1, 3, 5], 'cost': 243}, {'id': 18, 'elements': [5, 6, 9], 'cost': 228}, {'id': 19, 'elements': [10, 12, 13], 'cost': 207}, {'id': 20, 'elements': [11, 14, 15, 16, 18], 'cost': 260}, {'id': 21, 'elements': [16, 17], 'cost': 138}, {'id': 22, 'elements': [5, 11], 'cost': 38}, {'id': 23, 'elements': [7, 9], 'cost': 26}, {'id': 24, 'elements': [3, 7, 8], 'cost': 45}, {'id': 25, 'elements': [5, 7, 8, 10], 'cost': 380}, {'id': 26, 'elements': [9, 10, 11, 12], 'cost': 112}, {'id': 27, 'elements': [4, 5, 6, 7, 8], 'cost': 355}, {'id': 28, 'elements': [9, 14], 'cost': 126}, {'id': 29, 'elements': [1, 2], 'cost': 168}, {'id': 30, 'elements': [8, 9, 12], 'cost': 237}, {'id': 31, 'elements': [3, 6, 7, 8], 'cost': 156}, {'id': 32, 'elements': [1, 2, 6], 'cost': 39}, {'id': 33, 'elements': [14, 16, 17, 18], 'cost': 28}, {'id': 34, 'elements': [6, 8], 'cost': 122}, {'id': 35, 'elements': [3, 5, 6, 7], 'cost': 224}, {'id': 36, 'elements': [12, 16, 18], 'cost': 69}, {'id': 37, 'elements': [13, 14, 15, 16, 17], 'cost': 500}, {'id': 38, 'elements': [6, 9], 'cost': 52}, {'id': 39, 'elements': [5, 6, 8], 'cost': 168}, {'id': 40, 'elements': [13, 15], 'cost': 118}, {'id': 41, 'elements': [10, 11, 13, 15], 'cost': 32}, {'id': 42, 'elements': [10, 11, 12], 'cost': 237}, {'id': 43, 'elements': [14, 15, 16, 17, 18], 'cost': 380}, {'id': 44, 'elements': [13, 14, 16, 17], 'cost': 168}, {'id': 45, 'elements': [6, 8, 10], 'cost': 222}, {'id': 46, 'elements': [10, 11, 14, 15, 17], 'cost': 340}, {'id': 47, 'elements': [3, 4, 5], 'cost': 6}, {'id': 48, 'elements': [12, 13, 15, 17], 'cost': 48}, {'id': 49, 'elements': [8, 10], 'cost': 148}, {'id': 50, 'elements': [1, 2, 4, 5], 'cost': 96}, {'id': 51, 'elements': [6, 10, 11, 12, 13], 'cost': 445}, {'id': 52, 'elements': [2, 3, 4], 'cost': 237}, {'id': 53, 'elements': [4, 7, 8, 9], 'cost': 32}, {'id': 54, 'elements': [10, 12, 13, 14], 'cost': 188}, {'id': 55, 'elements': [16, 17, 18], 'cost': 54}, {'id': 56, 'elements': [6, 7, 8, 10], 'cost': 28}, {'id': 57, 'elements': [12, 13], 'cost': 32}, {'id': 58, 'elements': [15, 17, 18], 'cost': 240}, {'id': 59, 'elements': [1, 2, 5], 'cost': 153}, {'id': 60, 'elements': [7, 8, 9, 10, 13], 'cost': 100}, {'id': 61, 'elements': [13, 15, 16, 17, 18], 'cost': 305}, {'id': 62, 'elements': [11, 13, 16], 'cost': 9}, {'id': 63, 'elements': [6, 7, 8, 9, 10], 'cost': 210}, {'id': 64, 'elements': [14, 15, 16, 18], 'cost': 160}, {'id': 65, 'elements': [8, 9], 'cost': 74}, {'id': 66, 'elements': [1, 2, 3, 4, 6], 'cost': 345}, {'id': 67, 'elements': [6, 7, 8], 'cost': 60}, {'id': 68, 'elements': [1, 3], 'cost': 34}, {'id': 69, 'elements': [8, 10, 13], 'cost': 123}, {'id': 70, 'elements': [3, 4, 5, 6, 9], 'cost': 260}, {'id': 71, 'elements': [3, 5, 7, 8, 9], 'cost': 325}, {'id': 72, 'elements': [9, 10, 13], 'cost': 222}, {'id': 73, 'elements': [11, 13, 14, 16], 'cost': 348}, {'id': 74, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 75, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 76, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 77, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 78, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 79, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 80, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 81, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 82, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 83, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 84, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 85, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 86, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 87, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 88, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 89, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 90, 'elements': [17], 'cost': 10000, 'singleton': True}, {'id': 91, 'elements': [18], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0018_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0018_bag_stats.png'}","[1, 11, 33, 38, 41, 68]",224.0,"{'num_elements': 18, 'num_sets': 91, 'sets': [{'id': 'S1', 'elements': [4, 6, 7, 11], 'cost': 8}, {'id': 'S2', 'elements': [0, 1, 2, 3], 'cost': 116}, {'id': 'S3', 'elements': [3, 6, 7, 9], 'cost': 312}, {'id': 'S4', 'elements': [6, 7], 'cost': 66}, {'id': 'S5', 'elements': [1, 3, 5], 'cost': 165}, {'id': 'S6', 'elements': [7, 8, 9, 12], 'cost': 200}, {'id': 'S7', 'elements': [0, 1, 3], 'cost': 27}, {'id': 'S8', 'elements': [3, 4, 6, 7], 'cost': 396}, {'id': 'S9', 'elements': [1, 4, 5], 'cost': 30}, {'id': 'S10', 'elements': [4, 9], 'cost': 128}, {'id': 'S11', 'elements': [1, 3], 'cost': 70}, {'id': 'S12', 'elements': [10, 12], 'cost': 132}, {'id': 'S13', 'elements': [14, 15, 16, 17], 'cost': 36}, {'id': 'S14', 'elements': [8, 10, 11, 16], 'cost': 332}, {'id': 'S15', 'elements': [15, 17], 'cost': 118}, {'id': 'S16', 'elements': [6, 7, 8, 9, 10], 'cost': 450}, {'id': 'S17', 'elements': [0, 2, 4], 'cost': 243}, {'id': 'S18', 'elements': [4, 5, 8], 'cost': 228}, {'id': 'S19', 'elements': [9, 11, 12], 'cost': 207}, {'id': 'S20', 'elements': [10, 13, 14, 15, 17], 'cost': 260}, {'id': 'S21', 'elements': [15, 16], 'cost': 138}, {'id': 'S22', 'elements': [4, 10], 'cost': 38}, {'id': 'S23', 'elements': [6, 8], 'cost': 26}, {'id': 'S24', 'elements': [2, 6, 7], 'cost': 45}, {'id': 'S25', 'elements': [4, 6, 7, 9], 'cost': 380}, {'id': 'S26', 'elements': [8, 9, 10, 11], 'cost': 112}, {'id': 'S27', 'elements': [3, 4, 5, 6, 7], 'cost': 355}, {'id': 'S28', 'elements': [8, 13], 'cost': 126}, {'id': 'S29', 'elements': [0, 1], 'cost': 168}, {'id': 'S30', 'elements': [7, 8, 11], 'cost': 237}, {'id': 'S31', 'elements': [2, 5, 6, 7], 'cost': 156}, {'id': 'S32', 'elements': [0, 1, 5], 'cost': 39}, {'id': 'S33', 'elements': [13, 15, 16, 17], 'cost': 28}, {'id': 'S34', 'elements': [5, 7], 'cost': 122}, {'id': 'S35', 'elements': [2, 4, 5, 6], 'cost': 224}, {'id': 'S36', 'elements': [11, 15, 17], 'cost': 69}, {'id': 'S37', 'elements': [12, 13, 14, 15, 16], 'cost': 500}, {'id': 'S38', 'elements': [5, 8], 'cost': 52}, {'id': 'S39', 'elements': [4, 5, 7], 'cost': 168}, {'id': 'S40', 'elements': [12, 14], 'cost': 118}, {'id': 'S41', 'elements': [9, 10, 12, 14], 'cost': 32}, {'id': 'S42', 'elements': [9, 10, 11], 'cost': 237}, {'id': 'S43', 'elements': [13, 14, 15, 16, 17], 'cost': 380}, {'id': 'S44', 'elements': [12, 13, 15, 16], 'cost': 168}, {'id': 'S45', 'elements': [5, 7, 9], 'cost': 222}, {'id': 'S46', 'elements': [9, 10, 13, 14, 16], 'cost': 340}, {'id': 'S47', 'elements': [2, 3, 4], 'cost': 6}, {'id': 'S48', 'elements': [11, 12, 14, 16], 'cost': 48}, {'id': 'S49', 'elements': [7, 9], 'cost': 148}, {'id': 'S50', 'elements': [0, 1, 3, 4], 'cost': 96}, {'id': 'S51', 'elements': [5, 9, 10, 11, 12], 'cost': 445}, {'id': 'S52', 'elements': [1, 2, 3], 'cost': 237}, {'id': 'S53', 'elements': [3, 6, 7, 8], 'cost': 32}, {'id': 'S54', 'elements': [9, 11, 12, 13], 'cost': 188}, {'id': 'S55', 'elements': [15, 16, 17], 'cost': 54}, {'id': 'S56', 'elements': [5, 6, 7, 9], 'cost': 28}, {'id': 'S57', 'elements': [11, 12], 'cost': 32}, {'id': 'S58', 'elements': [14, 16, 17], 'cost': 240}, {'id': 'S59', 'elements': [0, 1, 4], 'cost': 153}, {'id': 'S60', 'elements': [6, 7, 8, 9, 12], 'cost': 100}, {'id': 'S61', 'elements': [12, 14, 15, 16, 17], 'cost': 305}, {'id': 'S62', 'elements': [10, 12, 15], 'cost': 9}, {'id': 'S63', 'elements': [5, 6, 7, 8, 9], 'cost': 210}, {'id': 'S64', 'elements': [13, 14, 15, 17], 'cost': 160}, {'id': 'S65', 'elements': [7, 8], 'cost': 74}, {'id': 'S66', 'elements': [0, 1, 2, 3, 5], 'cost': 345}, {'id': 'S67', 'elements': [5, 6, 7], 'cost': 60}, {'id': 'S68', 'elements': [0, 2], 'cost': 34}, {'id': 'S69', 'elements': [7, 9, 12], 'cost': 123}, {'id': 'S70', 'elements': [2, 3, 4, 5, 8], 'cost': 260}, {'id': 'S71', 'elements': [2, 4, 6, 7, 8], 'cost': 325}, {'id': 'S72', 'elements': [8, 9, 12], 'cost': 222}, {'id': 'S73', 'elements': [10, 12, 13, 15], 'cost': 348}, {'id': 'S74', 'elements': [0], 'cost': 10000}, {'id': 'S75', 'elements': [1], 'cost': 10000}, {'id': 'S76', 'elements': [2], 'cost': 10000}, {'id': 'S77', 'elements': [3], 'cost': 10000}, {'id': 'S78', 'elements': [4], 'cost': 10000}, {'id': 'S79', 'elements': [5], 'cost': 10000}, {'id': 'S80', 'elements': [6], 'cost': 10000}, {'id': 'S81', 'elements': [7], 'cost': 10000}, {'id': 'S82', 'elements': [8], 'cost': 10000}, {'id': 'S83', 'elements': [9], 'cost': 10000}, {'id': 'S84', 'elements': [10], 'cost': 10000}, {'id': 'S85', 'elements': [11], 'cost': 10000}, {'id': 'S86', 'elements': [12], 'cost': 10000}, {'id': 'S87', 'elements': [13], 'cost': 10000}, {'id': 'S88', 'elements': [14], 'cost': 10000}, {'id': 'S89', 'elements': [15], 'cost': 10000}, {'id': 'S90', 'elements': [16], 'cost': 10000}, {'id': 'S91', 'elements': [17], 'cost': 10000}]}","['S1', 'S11', 'S33', 'S38', 'S41', 'S68']",19,json,0
SPP,SPP,"Back when a couple downsized, they faced a pile of belongings and a handful of potential crates and had to pick which crates to use so every item got packed into one chosen crate with nothing omitted or doubled up. Since every crate costs money, the winning plan was the one with the lowest total cost, calculated by adding the prices of the selected crates. The item list and crate options appear below.

- **total_items_count**: 17
- **total_available_crates**: 67

| crate_id | crate_cost | items_in_crate |
|---|---|---|
| S1 | 16 | 5 7 |
| S2 | 12 | 4 5 6 9 |
| S3 | 132 | 15 16 |
| S4 | 258 | 14 15 16 |
| S5 | 6 | 3 4 5 |
| S6 | 84 | 10 11 12 |
| S7 | 376 | 11 13 14 15 |
| S8 | 195 | 10 12 13 14 15 |
| S9 | 291 | 13 15 16 |
| S10 | 216 | 0 1 3 |
| S11 | 189 | 3 4 6 |
| S12 | 144 | 8 11 |
| S13 | 328 | 2 4 6 10 |
| S14 | 96 | 8 9 11 |
| S15 | 244 | 10 11 12 13 |
| S16 | 95 | 0 2 3 4 5 |
| S17 | 246 | 1 3 4 |
| S18 | 405 | 8 9 10 11 12 |
| S19 | 84 | 1 3 |
| S20 | 95 | 7 8 9 10 11 |
| S21 | 106 | 14 16 |
| S22 | 12 | 1 2 3 6 |
| S23 | 231 | 7 9 10 |
| S24 | 48 | 8 10 12 |
| S25 | 54 | 3 5 |
| S26 | 30 | 2 4 |
| S27 | 28 | 12 15 |
| S28 | 135 | 0 1 2 3 4 |
| S29 | 144 | 3 4 6 8 |
| S30 | 45 | 9 10 11 12 13 |
| S31 | 74 | 1 2 |
| S32 | 56 | 0 4 |
| S33 | 230 | 11 13 14 15 16 |
| S34 | 75 | 10 11 12 13 16 |
| S35 | 118 | 9 10 |
| S36 | 220 | 7 8 10 11 |
| S37 | 174 | 3 4 |
| S38 | 171 | 0 1 5 |
| S39 | 42 | 0 2 4 |
| S40 | 208 | 2 4 6 8 |
| S41 | 184 | 12 14 |
| S42 | 344 | 7 9 10 11 |
| S43 | 64 | 10 12 13 16 |
| S44 | 285 | 11 14 16 |
| S45 | 46 | 0 2 |
| S46 | 30 | 7 8 11 |
| S47 | 252 | 13 14 15 16 |
| S48 | 294 | 6 10 11 |
| S49 | 116 | 11 15 |
| S50 | 255 | 9 10 12 |
| S51 | 10000 | 0 |
| S52 | 10000 | 1 |
| S53 | 10000 | 2 |
| S54 | 10000 | 3 |
| S55 | 10000 | 4 |
| S56 | 10000 | 5 |
| S57 | 10000 | 6 |
| S58 | 10000 | 7 |
| S59 | 10000 | 8 |
| S60 | 10000 | 9 |
| S61 | 10000 | 10 |
| S62 | 10000 | 11 |
| S63 | 10000 | 12 |
| S64 | 10000 | 13 |
| S65 | 10000 | 14 |
| S66 | 10000 | 15 |
| S67 | 10000 | 16 |

If you want to tell me which crates you’d pick, just use this little JSON shape so I can read the chosen crate IDs cleanly:

{
  ""solution"": [""crate_id"", ...]
}

Think of it like a checklist: ""solution"" is the list of crates you picked (put the exact crate identifiers from the instance in that array). I used the placeholder crate_id to match the packing story — swap that out for the real IDs when you reply. This JSON is just a sketch of the shape I expect, not the actual answer.

Please be sure to use the identifiers exactly as they appear in the instance input — no renaming and no new labels. 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 17, 'num_sets': 67, 'density': 0.15715539947322213, 'sets': [{'id': 1, 'elements': [6, 8], 'cost': 16}, {'id': 2, 'elements': [5, 6, 7, 10], 'cost': 12}, {'id': 3, 'elements': [16, 17], 'cost': 132}, {'id': 4, 'elements': [15, 16, 17], 'cost': 258}, {'id': 5, 'elements': [4, 5, 6], 'cost': 6}, {'id': 6, 'elements': [11, 12, 13], 'cost': 84}, {'id': 7, 'elements': [12, 14, 15, 16], 'cost': 376}, {'id': 8, 'elements': [11, 13, 14, 15, 16], 'cost': 195}, {'id': 9, 'elements': [14, 16, 17], 'cost': 291}, {'id': 10, 'elements': [1, 2, 4], 'cost': 216}, {'id': 11, 'elements': [4, 5, 7], 'cost': 189}, {'id': 12, 'elements': [9, 12], 'cost': 144}, {'id': 13, 'elements': [3, 5, 7, 11], 'cost': 328}, {'id': 14, 'elements': [9, 10, 12], 'cost': 96}, {'id': 15, 'elements': [11, 12, 13, 14], 'cost': 244}, {'id': 16, 'elements': [1, 3, 4, 5, 6], 'cost': 95}, {'id': 17, 'elements': [2, 4, 5], 'cost': 246}, {'id': 18, 'elements': [9, 10, 11, 12, 13], 'cost': 405}, {'id': 19, 'elements': [2, 4], 'cost': 84}, {'id': 20, 'elements': [8, 9, 10, 11, 12], 'cost': 95}, {'id': 21, 'elements': [15, 17], 'cost': 106}, {'id': 22, 'elements': [2, 3, 4, 7], 'cost': 12}, {'id': 23, 'elements': [8, 10, 11], 'cost': 231}, {'id': 24, 'elements': [9, 11, 13], 'cost': 48}, {'id': 25, 'elements': [4, 6], 'cost': 54}, {'id': 26, 'elements': [3, 5], 'cost': 30}, {'id': 27, 'elements': [13, 16], 'cost': 28}, {'id': 28, 'elements': [1, 2, 3, 4, 5], 'cost': 135}, {'id': 29, 'elements': [4, 5, 7, 9], 'cost': 144}, {'id': 30, 'elements': [10, 11, 12, 13, 14], 'cost': 45}, {'id': 31, 'elements': [2, 3], 'cost': 74}, {'id': 32, 'elements': [1, 5], 'cost': 56}, {'id': 33, 'elements': [12, 14, 15, 16, 17], 'cost': 230}, {'id': 34, 'elements': [11, 12, 13, 14, 17], 'cost': 75}, {'id': 35, 'elements': [10, 11], 'cost': 118}, {'id': 36, 'elements': [8, 9, 11, 12], 'cost': 220}, {'id': 37, 'elements': [4, 5], 'cost': 174}, {'id': 38, 'elements': [1, 2, 6], 'cost': 171}, {'id': 39, 'elements': [1, 3, 5], 'cost': 42}, {'id': 40, 'elements': [3, 5, 7, 9], 'cost': 208}, {'id': 41, 'elements': [13, 15], 'cost': 184}, {'id': 42, 'elements': [8, 10, 11, 12], 'cost': 344}, {'id': 43, 'elements': [11, 13, 14, 17], 'cost': 64}, {'id': 44, 'elements': [12, 15, 17], 'cost': 285}, {'id': 45, 'elements': [1, 3], 'cost': 46}, {'id': 46, 'elements': [8, 9, 12], 'cost': 30}, {'id': 47, 'elements': [14, 15, 16, 17], 'cost': 252}, {'id': 48, 'elements': [7, 11, 12], 'cost': 294}, {'id': 49, 'elements': [12, 16], 'cost': 116}, {'id': 50, 'elements': [10, 11, 13], 'cost': 255}, {'id': 51, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 66, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 67, 'elements': [17], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0019_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0019_bag_stats.png'}","[1, 12, 22, 32, 47, 50]",735.0,"{'num_elements': 17, 'num_sets': 67, 'sets': [{'id': 'S1', 'elements': [5, 7], 'cost': 16}, {'id': 'S2', 'elements': [4, 5, 6, 9], 'cost': 12}, {'id': 'S3', 'elements': [15, 16], 'cost': 132}, {'id': 'S4', 'elements': [14, 15, 16], 'cost': 258}, {'id': 'S5', 'elements': [3, 4, 5], 'cost': 6}, {'id': 'S6', 'elements': [10, 11, 12], 'cost': 84}, {'id': 'S7', 'elements': [11, 13, 14, 15], 'cost': 376}, {'id': 'S8', 'elements': [10, 12, 13, 14, 15], 'cost': 195}, {'id': 'S9', 'elements': [13, 15, 16], 'cost': 291}, {'id': 'S10', 'elements': [0, 1, 3], 'cost': 216}, {'id': 'S11', 'elements': [3, 4, 6], 'cost': 189}, {'id': 'S12', 'elements': [8, 11], 'cost': 144}, {'id': 'S13', 'elements': [2, 4, 6, 10], 'cost': 328}, {'id': 'S14', 'elements': [8, 9, 11], 'cost': 96}, {'id': 'S15', 'elements': [10, 11, 12, 13], 'cost': 244}, {'id': 'S16', 'elements': [0, 2, 3, 4, 5], 'cost': 95}, {'id': 'S17', 'elements': [1, 3, 4], 'cost': 246}, {'id': 'S18', 'elements': [8, 9, 10, 11, 12], 'cost': 405}, {'id': 'S19', 'elements': [1, 3], 'cost': 84}, {'id': 'S20', 'elements': [7, 8, 9, 10, 11], 'cost': 95}, {'id': 'S21', 'elements': [14, 16], 'cost': 106}, {'id': 'S22', 'elements': [1, 2, 3, 6], 'cost': 12}, {'id': 'S23', 'elements': [7, 9, 10], 'cost': 231}, {'id': 'S24', 'elements': [8, 10, 12], 'cost': 48}, {'id': 'S25', 'elements': [3, 5], 'cost': 54}, {'id': 'S26', 'elements': [2, 4], 'cost': 30}, {'id': 'S27', 'elements': [12, 15], 'cost': 28}, {'id': 'S28', 'elements': [0, 1, 2, 3, 4], 'cost': 135}, {'id': 'S29', 'elements': [3, 4, 6, 8], 'cost': 144}, {'id': 'S30', 'elements': [9, 10, 11, 12, 13], 'cost': 45}, {'id': 'S31', 'elements': [1, 2], 'cost': 74}, {'id': 'S32', 'elements': [0, 4], 'cost': 56}, {'id': 'S33', 'elements': [11, 13, 14, 15, 16], 'cost': 230}, {'id': 'S34', 'elements': [10, 11, 12, 13, 16], 'cost': 75}, {'id': 'S35', 'elements': [9, 10], 'cost': 118}, {'id': 'S36', 'elements': [7, 8, 10, 11], 'cost': 220}, {'id': 'S37', 'elements': [3, 4], 'cost': 174}, {'id': 'S38', 'elements': [0, 1, 5], 'cost': 171}, {'id': 'S39', 'elements': [0, 2, 4], 'cost': 42}, {'id': 'S40', 'elements': [2, 4, 6, 8], 'cost': 208}, {'id': 'S41', 'elements': [12, 14], 'cost': 184}, {'id': 'S42', 'elements': [7, 9, 10, 11], 'cost': 344}, {'id': 'S43', 'elements': [10, 12, 13, 16], 'cost': 64}, {'id': 'S44', 'elements': [11, 14, 16], 'cost': 285}, {'id': 'S45', 'elements': [0, 2], 'cost': 46}, {'id': 'S46', 'elements': [7, 8, 11], 'cost': 30}, {'id': 'S47', 'elements': [13, 14, 15, 16], 'cost': 252}, {'id': 'S48', 'elements': [6, 10, 11], 'cost': 294}, {'id': 'S49', 'elements': [11, 15], 'cost': 116}, {'id': 'S50', 'elements': [9, 10, 12], 'cost': 255}, {'id': 'S51', 'elements': [0], 'cost': 10000}, {'id': 'S52', 'elements': [1], 'cost': 10000}, {'id': 'S53', 'elements': [2], 'cost': 10000}, {'id': 'S54', 'elements': [3], 'cost': 10000}, {'id': 'S55', 'elements': [4], 'cost': 10000}, {'id': 'S56', 'elements': [5], 'cost': 10000}, {'id': 'S57', 'elements': [6], 'cost': 10000}, {'id': 'S58', 'elements': [7], 'cost': 10000}, {'id': 'S59', 'elements': [8], 'cost': 10000}, {'id': 'S60', 'elements': [9], 'cost': 10000}, {'id': 'S61', 'elements': [10], 'cost': 10000}, {'id': 'S62', 'elements': [11], 'cost': 10000}, {'id': 'S63', 'elements': [12], 'cost': 10000}, {'id': 'S64', 'elements': [13], 'cost': 10000}, {'id': 'S65', 'elements': [14], 'cost': 10000}, {'id': 'S66', 'elements': [15], 'cost': 10000}, {'id': 'S67', 'elements': [16], 'cost': 10000}]}","['S1', 'S12', 'S22', 'S32', 'S47', 'S50']",20,markdown_table,0
SPP,SPP,"I was putting together an event and had a stack of vendor packages to choose from — each package covers some of the planned activities and has a price. The job is to pick a handful of those packages so every activity on the schedule is handled by one and only one package (nothing left out, nothing covered twice), and to keep the total bill as small as possible by adding up the prices of the chosen packages. The concrete packages and the activity list are shown below.

{
  ""total_activities"": 15,
  ""total_packages"": 72,
  ""sets"": [
    {
      ""package_id"": ""S1"",
      ""covered_activities"": [
        ""E"",
        ""H""
      ],
      ""package_price"": 126
    },
    {
      ""package_id"": ""S2"",
      ""covered_activities"": [
        ""I"",
        ""J"",
        ""K"",
        ""L""
      ],
      ""package_price"": 244
    },
    {
      ""package_id"": ""S3"",
      ""covered_activities"": [
        ""B"",
        ""C"",
        ""D""
      ],
      ""package_price"": 147
    },
    {
      ""package_id"": ""S4"",
      ""covered_activities"": [
        ""C"",
        ""E"",
        ""G""
      ],
      ""package_price"": 228
    },
    {
      ""package_id"": ""S5"",
      ""covered_activities"": [
        ""M"",
        ""O""
      ],
      ""package_price"": 150
    },
    {
      ""package_id"": ""S6"",
      ""covered_activities"": [
        ""C"",
        ""E"",
        ""F""
      ],
      ""package_price"": 93
    },
    {
      ""package_id"": ""S7"",
      ""covered_activities"": [
        ""H"",
        ""K"",
        ""L""
      ],
      ""package_price"": 45
    },
    {
      ""package_id"": ""S8"",
      ""covered_activities"": [
        ""A"",
        ""B"",
        ""C""
      ],
      ""package_price"": 165
    },
    {
      ""package_id"": ""S9"",
      ""covered_activities"": [
        ""K"",
        ""N"",
        ""O""
      ],
      ""package_price"": 9
    },
    {
      ""package_id"": ""S10"",
      ""covered_activities"": [
        ""E"",
        ""F"",
        ""G"",
        ""H""
      ],
      ""package_price"": 356
    },
    {
      ""package_id"": ""S11"",
      ""covered_activities"": [
        ""D"",
        ""F"",
        ""G"",
        ""H""
      ],
      ""package_price"": 156
    },
    {
      ""package_id"": ""S12"",
      ""covered_activities"": [
        ""F"",
        ""H"",
        ""J""
      ],
      ""package_price"": 30
    },
    {
      ""package_id"": ""S13"",
      ""covered_activities"": [
        ""H"",
        ""I"",
        ""J"",
        ""K""
      ],
      ""package_price"": 116
    },
    {
      ""package_id"": ""S14"",
      ""covered_activities"": [
        ""J"",
        ""K""
      ],
      ""package_price"": 54
    },
    {
      ""package_id"": ""S15"",
      ""covered_activities"": [
        ""A"",
        ""C"",
        ""D""
      ],
      ""package_price"": 168
    },
    {
      ""package_id"": ""S16"",
      ""covered_activities"": [
        ""E"",
        ""F"",
        ""G""
      ],
      ""package_price"": 177
    },
    {
      ""package_id"": ""S17"",
      ""covered_activities"": [
        ""F"",
        ""H"",
        ""J"",
        ""K""
      ],
      ""package_price"": 296
    },
    {
      ""package_id"": ""S18"",
      ""covered_activities"": [
        ""J"",
        ""L"",
        ""N""
      ],
      ""package_price"": 246
    },
    {
      ""package_id"": ""S19"",
      ""covered_activities"": [
        ""K"",
        ""L"",
        ""M""
      ],
      ""package_price"": 282
    },
    {
      ""package_id"": ""S20"",
      ""covered_activities"": [
        ""G"",
        ""H"",
        ""I""
      ],
      ""package_price"": 288
    },
    {
      ""package_id"": ""S21"",
      ""covered_activities"": [
        ""M"",
        ""N"",
        ""O""
      ],
      ""package_price"": 99
    },
    {
      ""package_id"": ""S22"",
      ""covered_activities"": [
        ""B"",
        ""C"",
        ""D"",
        ""E""
      ],
      ""package_price"": 160
    },
    {
      ""package_id"": ""S23"",
      ""covered_activities"": [
        ""I"",
        ""K"",
        ""M""
      ],
      ""package_price"": 78
    },
    {
      ""package_id"": ""S24"",
      ""covered_activities"": [
        ""A"",
        ""D""
      ],
      ""package_price"": 72
    },
    {
      ""package_id"": ""S25"",
      ""covered_activities"": [
        ""A"",
        ""B"",
        ""C"",
        ""E""
      ],
      ""package_price"": 212
    },
    {
      ""package_id"": ""S26"",
      ""covered_activities"": [
        ""I"",
        ""K"",
        ""L"",
        ""N""
      ],
      ""package_price"": 72
    },
    {
      ""package_id"": ""S27"",
      ""covered_activities"": [
        ""K"",
        ""L""
      ],
      ""package_price"": 134
    },
    {
      ""package_id"": ""S28"",
      ""covered_activities"": [
        ""I"",
        ""K"",
        ""L""
      ],
      ""package_price"": 36
    },
    {
      ""package_id"": ""S29"",
      ""covered_activities"": [
        ""A"",
        ""C""
      ],
      ""package_price"": 150
    },
    {
      ""package_id"": ""S30"",
      ""covered_activities"": [
        ""J"",
        ""L"",
        ""M"",
        ""N""
      ],
      ""package_price"": 92
    },
    {
      ""package_id"": ""S31"",
      ""covered_activities"": [
        ""D"",
        ""E""
      ],
      ""package_price"": 152
    },
    {
      ""package_id"": ""S32"",
      ""covered_activities"": [
        ""I"",
        ""L""
      ],
      ""package_price"": 82
    },
    {
      ""package_id"": ""S33"",
      ""covered_activities"": [
        ""I"",
        ""J"",
        ""K""
      ],
      ""package_price"": 57
    },
    {
      ""package_id"": ""S34"",
      ""covered_activities"": [
        ""A"",
        ""B"",
        ""C"",
        ""D""
      ],
      ""package_price"": 344
    },
    {
      ""package_id"": ""S35"",
      ""covered_activities"": [
        ""I"",
        ""J""
      ],
      ""package_price"": 180
    },
    {
      ""package_id"": ""S36"",
      ""covered_activities"": [
        ""K"",
        ""M"",
        ""N""
      ],
      ""package_price"": 171
    },
    {
      ""package_id"": ""S37"",
      ""covered_activities"": [
        ""M"",
        ""N""
      ],
      ""package_price"": 82
    },
    {
      ""package_id"": ""S38"",
      ""covered_activities"": [
        ""J"",
        ""K"",
        ""M"",
        ""N""
      ],
      ""package_price"": 400
    },
    {
      ""package_id"": ""S39"",
      ""covered_activities"": [
        ""G"",
        ""H"",
        ""I"",
        ""J""
      ],
      ""package_price"": 52
    },
    {
      ""package_id"": ""S40"",
      ""covered_activities"": [
        ""J"",
        ""K"",
        ""L"",
        ""N""
      ],
      ""package_price"": 284
    },
    {
      ""package_id"": ""S41"",
      ""covered_activities"": [
        ""F"",
        ""G"",
        ""H"",
        ""I""
      ],
      ""package_price"": 228
    },
    {
      ""package_id"": ""S42"",
      ""covered_activities"": [
        ""D"",
        ""G""
      ],
      ""package_price"": 168
    },
    {
      ""package_id"": ""S43"",
      ""covered_activities"": [
        ""L"",
        ""M"",
        ""N"",
        ""O""
      ],
      ""package_price"": 380
    },
    {
      ""package_id"": ""S44"",
      ""covered_activities"": [
        ""I"",
        ""J"",
        ""L"",
        ""M""
      ],
      ""package_price"": 324
    },
    {
      ""package_id"": ""S45"",
      ""covered_activities"": [
        ""A"",
        ""E""
      ],
      ""package_price"": 180
    },
    {
      ""package_id"": ""S46"",
      ""covered_activities"": [
        ""H"",
        ""J"",
        ""K""
      ],
      ""package_price"": 177
    },
    {
      ""package_id"": ""S47"",
      ""covered_activities"": [
        ""F"",
        ""G"",
        ""I"",
        ""J""
      ],
      ""package_price"": 140
    },
    {
      ""package_id"": ""S48"",
      ""covered_activities"": [
        ""G"",
        ""H""
      ],
      ""package_price"": 54
    },
    {
      ""package_id"": ""S49"",
      ""covered_activities"": [
        ""A"",
        ""B"",
        ""F""
      ],
      ""package_price"": 174
    },
    {
      ""package_id"": ""S50"",
      ""covered_activities"": [
        ""D"",
        ""E"",
        ""F"",
        ""G""
      ],
      ""package_price"": 168
    },
    {
      ""package_id"": ""S51"",
      ""covered_activities"": [
        ""A"",
        ""B""
      ],
      ""package_price"": 62
    },
    {
      ""package_id"": ""S52"",
      ""covered_activities"": [
        ""H"",
        ""I"",
        ""J""
      ],
      ""package_price"": 231
    },
    {
      ""package_id"": ""S53"",
      ""covered_activities"": [
        ""C"",
        ""D"",
        ""H""
      ],
      ""package_price"": 69
    },
    {
      ""package_id"": ""S54"",
      ""covered_activities"": [
        ""H"",
        ""I""
      ],
      ""package_price"": 120
    },
    {
      ""package_id"": ""S55"",
      ""covered_activities"": [
        ""E"",
        ""G""
      ],
      ""package_price"": 68
    },
    {
      ""package_id"": ""S56"",
      ""covered_activities"": [
        ""B"",
        ""D""
      ],
      ""package_price"": 72
    },
    {
      ""package_id"": ""S57"",
      ""covered_activities"": [
        ""H"",
        ""I"",
        ""K"",
        ""M""
      ],
      ""package_price"": 392
    },
    {
      ""package_id"": ""S58"",
      ""covered_activities"": [
        ""A""
      ],
      ""package_price"": 10000
    },
    {
      ""package_id"": ""S59"",
      ""covered_activities"": [
        ""B""
      ],
      ""package_price"": 10000
    },
    {
      ""package_id"": ""S60"",
      ""covered_activities"": [
        ""C""
      ],
      ""package_price"": 10000
    },
    {
      ""package_id"": ""S61"",
      ""covered_activities"": [
        ""D""
      ],
      ""package_price"": 10000
    },
    {
      ""package_id"": ""S62"",
      ""covered_activities"": [
        ""E""
      ],
      ""package_price"": 10000
    },
    {
      ""package_id"": ""S63"",
      ""covered_activities"": [
        ""F""
      ],
      ""package_price"": 10000
    },
    {
      ""package_id"": ""S64"",
      ""covered_activities"": [
        ""G""
      ],
      ""package_price"": 10000
    },
    {
      ""package_id"": ""S65"",
      ""covered_activities"": [
        ""H""
      ],
      ""package_price"": 10000
    },
    {
      ""package_id"": ""S66"",
      ""covered_activities"": [
        ""I""
      ],
      ""package_price"": 10000
    },
    {
      ""package_id"": ""S67"",
      ""covered_activities"": [
        ""J""
      ],
      ""package_price"": 10000
    },
    {
      ""package_id"": ""S68"",
      ""covered_activities"": [
        ""K""
      ],
      ""package_price"": 10000
    },
    {
      ""package_id"": ""S69"",
      ""covered_activities"": [
        ""L""
      ],
      ""package_price"": 10000
    },
    {
      ""package_id"": ""S70"",
      ""covered_activities"": [
        ""M""
      ],
      ""package_price"": 10000
    },
    {
      ""package_id"": ""S71"",
      ""covered_activities"": [
        ""N""
      ],
      ""package_price"": 10000
    },
    {
      ""package_id"": ""S72"",
      ""covered_activities"": [
        ""O""
      ],
      ""package_price"": 10000
    }
  ]
}

If you want to send the chosen packages back to me, just drop them in a little JSON snippet like this so it's easy to parse:

{
  ""solution"": [""package_id"", ...]
}

Think of ""solution"" as the list of package IDs you picked. ""package_id"" is just a placeholder showing the kind of entries I expect — replace it with the actual package identifiers from the problem (that could be numbers, letters, or mixes). This is just a sketch of the shape I want, not the final answer itself.

Please make sure to use the exact identifiers from the instance input — do not rename them or invent new labels.  
For example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 15, 'num_sets': 72, 'density': 0.17407407407407408, 'sets': [{'id': 1, 'elements': [5, 8], 'cost': 126}, {'id': 2, 'elements': [9, 10, 11, 12], 'cost': 244}, {'id': 3, 'elements': [2, 3, 4], 'cost': 147}, {'id': 4, 'elements': [3, 5, 7], 'cost': 228}, {'id': 5, 'elements': [13, 15], 'cost': 150}, {'id': 6, 'elements': [3, 5, 6], 'cost': 93}, {'id': 7, 'elements': [8, 11, 12], 'cost': 45}, {'id': 8, 'elements': [1, 2, 3], 'cost': 165}, {'id': 9, 'elements': [11, 14, 15], 'cost': 9}, {'id': 10, 'elements': [5, 6, 7, 8], 'cost': 356}, {'id': 11, 'elements': [4, 6, 7, 8], 'cost': 156}, {'id': 12, 'elements': [6, 8, 10], 'cost': 30}, {'id': 13, 'elements': [8, 9, 10, 11], 'cost': 116}, {'id': 14, 'elements': [10, 11], 'cost': 54}, {'id': 15, 'elements': [1, 3, 4], 'cost': 168}, {'id': 16, 'elements': [5, 6, 7], 'cost': 177}, {'id': 17, 'elements': [6, 8, 10, 11], 'cost': 296}, {'id': 18, 'elements': [10, 12, 14], 'cost': 246}, {'id': 19, 'elements': [11, 12, 13], 'cost': 282}, {'id': 20, 'elements': [7, 8, 9], 'cost': 288}, {'id': 21, 'elements': [13, 14, 15], 'cost': 99}, {'id': 22, 'elements': [2, 3, 4, 5], 'cost': 160}, {'id': 23, 'elements': [9, 11, 13], 'cost': 78}, {'id': 24, 'elements': [1, 4], 'cost': 72}, {'id': 25, 'elements': [1, 2, 3, 5], 'cost': 212}, {'id': 26, 'elements': [9, 11, 12, 14], 'cost': 72}, {'id': 27, 'elements': [11, 12], 'cost': 134}, {'id': 28, 'elements': [9, 11, 12], 'cost': 36}, {'id': 29, 'elements': [1, 3], 'cost': 150}, {'id': 30, 'elements': [10, 12, 13, 14], 'cost': 92}, {'id': 31, 'elements': [4, 5], 'cost': 152}, {'id': 32, 'elements': [9, 12], 'cost': 82}, {'id': 33, 'elements': [9, 10, 11], 'cost': 57}, {'id': 34, 'elements': [1, 2, 3, 4], 'cost': 344}, {'id': 35, 'elements': [9, 10], 'cost': 180}, {'id': 36, 'elements': [11, 13, 14], 'cost': 171}, {'id': 37, 'elements': [13, 14], 'cost': 82}, {'id': 38, 'elements': [10, 11, 13, 14], 'cost': 400}, {'id': 39, 'elements': [7, 8, 9, 10], 'cost': 52}, {'id': 40, 'elements': [10, 11, 12, 14], 'cost': 284}, {'id': 41, 'elements': [6, 7, 8, 9], 'cost': 228}, {'id': 42, 'elements': [4, 7], 'cost': 168}, {'id': 43, 'elements': [12, 13, 14, 15], 'cost': 380}, {'id': 44, 'elements': [9, 10, 12, 13], 'cost': 324}, {'id': 45, 'elements': [1, 5], 'cost': 180}, {'id': 46, 'elements': [8, 10, 11], 'cost': 177}, {'id': 47, 'elements': [6, 7, 9, 10], 'cost': 140}, {'id': 48, 'elements': [7, 8], 'cost': 54}, {'id': 49, 'elements': [1, 2, 6], 'cost': 174}, {'id': 50, 'elements': [4, 5, 6, 7], 'cost': 168}, {'id': 51, 'elements': [1, 2], 'cost': 62}, {'id': 52, 'elements': [8, 9, 10], 'cost': 231}, {'id': 53, 'elements': [3, 4, 8], 'cost': 69}, {'id': 54, 'elements': [8, 9], 'cost': 120}, {'id': 55, 'elements': [5, 7], 'cost': 68}, {'id': 56, 'elements': [2, 4], 'cost': 72}, {'id': 57, 'elements': [8, 9, 11, 13], 'cost': 392}, {'id': 58, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 66, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 67, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 68, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 69, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 70, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 71, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 72, 'elements': [15], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0020_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0020_bag_stats.png'}","[12, 21, 28, 29, 55, 56]",455.0,"{'num_elements': 15, 'num_sets': 72, 'sets': [{'id': 'S1', 'elements': ['E', 'H'], 'cost': 126}, {'id': 'S2', 'elements': ['I', 'J', 'K', 'L'], 'cost': 244}, {'id': 'S3', 'elements': ['B', 'C', 'D'], 'cost': 147}, {'id': 'S4', 'elements': ['C', 'E', 'G'], 'cost': 228}, {'id': 'S5', 'elements': ['M', 'O'], 'cost': 150}, {'id': 'S6', 'elements': ['C', 'E', 'F'], 'cost': 93}, {'id': 'S7', 'elements': ['H', 'K', 'L'], 'cost': 45}, {'id': 'S8', 'elements': ['A', 'B', 'C'], 'cost': 165}, {'id': 'S9', 'elements': ['K', 'N', 'O'], 'cost': 9}, {'id': 'S10', 'elements': ['E', 'F', 'G', 'H'], 'cost': 356}, {'id': 'S11', 'elements': ['D', 'F', 'G', 'H'], 'cost': 156}, {'id': 'S12', 'elements': ['F', 'H', 'J'], 'cost': 30}, {'id': 'S13', 'elements': ['H', 'I', 'J', 'K'], 'cost': 116}, {'id': 'S14', 'elements': ['J', 'K'], 'cost': 54}, {'id': 'S15', 'elements': ['A', 'C', 'D'], 'cost': 168}, {'id': 'S16', 'elements': ['E', 'F', 'G'], 'cost': 177}, {'id': 'S17', 'elements': ['F', 'H', 'J', 'K'], 'cost': 296}, {'id': 'S18', 'elements': ['J', 'L', 'N'], 'cost': 246}, {'id': 'S19', 'elements': ['K', 'L', 'M'], 'cost': 282}, {'id': 'S20', 'elements': ['G', 'H', 'I'], 'cost': 288}, {'id': 'S21', 'elements': ['M', 'N', 'O'], 'cost': 99}, {'id': 'S22', 'elements': ['B', 'C', 'D', 'E'], 'cost': 160}, {'id': 'S23', 'elements': ['I', 'K', 'M'], 'cost': 78}, {'id': 'S24', 'elements': ['A', 'D'], 'cost': 72}, {'id': 'S25', 'elements': ['A', 'B', 'C', 'E'], 'cost': 212}, {'id': 'S26', 'elements': ['I', 'K', 'L', 'N'], 'cost': 72}, {'id': 'S27', 'elements': ['K', 'L'], 'cost': 134}, {'id': 'S28', 'elements': ['I', 'K', 'L'], 'cost': 36}, {'id': 'S29', 'elements': ['A', 'C'], 'cost': 150}, {'id': 'S30', 'elements': ['J', 'L', 'M', 'N'], 'cost': 92}, {'id': 'S31', 'elements': ['D', 'E'], 'cost': 152}, {'id': 'S32', 'elements': ['I', 'L'], 'cost': 82}, {'id': 'S33', 'elements': ['I', 'J', 'K'], 'cost': 57}, {'id': 'S34', 'elements': ['A', 'B', 'C', 'D'], 'cost': 344}, {'id': 'S35', 'elements': ['I', 'J'], 'cost': 180}, {'id': 'S36', 'elements': ['K', 'M', 'N'], 'cost': 171}, {'id': 'S37', 'elements': ['M', 'N'], 'cost': 82}, {'id': 'S38', 'elements': ['J', 'K', 'M', 'N'], 'cost': 400}, {'id': 'S39', 'elements': ['G', 'H', 'I', 'J'], 'cost': 52}, {'id': 'S40', 'elements': ['J', 'K', 'L', 'N'], 'cost': 284}, {'id': 'S41', 'elements': ['F', 'G', 'H', 'I'], 'cost': 228}, {'id': 'S42', 'elements': ['D', 'G'], 'cost': 168}, {'id': 'S43', 'elements': ['L', 'M', 'N', 'O'], 'cost': 380}, {'id': 'S44', 'elements': ['I', 'J', 'L', 'M'], 'cost': 324}, {'id': 'S45', 'elements': ['A', 'E'], 'cost': 180}, {'id': 'S46', 'elements': ['H', 'J', 'K'], 'cost': 177}, {'id': 'S47', 'elements': ['F', 'G', 'I', 'J'], 'cost': 140}, {'id': 'S48', 'elements': ['G', 'H'], 'cost': 54}, {'id': 'S49', 'elements': ['A', 'B', 'F'], 'cost': 174}, {'id': 'S50', 'elements': ['D', 'E', 'F', 'G'], 'cost': 168}, {'id': 'S51', 'elements': ['A', 'B'], 'cost': 62}, {'id': 'S52', 'elements': ['H', 'I', 'J'], 'cost': 231}, {'id': 'S53', 'elements': ['C', 'D', 'H'], 'cost': 69}, {'id': 'S54', 'elements': ['H', 'I'], 'cost': 120}, {'id': 'S55', 'elements': ['E', 'G'], 'cost': 68}, {'id': 'S56', 'elements': ['B', 'D'], 'cost': 72}, {'id': 'S57', 'elements': ['H', 'I', 'K', 'M'], 'cost': 392}, {'id': 'S58', 'elements': ['A'], 'cost': 10000}, {'id': 'S59', 'elements': ['B'], 'cost': 10000}, {'id': 'S60', 'elements': ['C'], 'cost': 10000}, {'id': 'S61', 'elements': ['D'], 'cost': 10000}, {'id': 'S62', 'elements': ['E'], 'cost': 10000}, {'id': 'S63', 'elements': ['F'], 'cost': 10000}, {'id': 'S64', 'elements': ['G'], 'cost': 10000}, {'id': 'S65', 'elements': ['H'], 'cost': 10000}, {'id': 'S66', 'elements': ['I'], 'cost': 10000}, {'id': 'S67', 'elements': ['J'], 'cost': 10000}, {'id': 'S68', 'elements': ['K'], 'cost': 10000}, {'id': 'S69', 'elements': ['L'], 'cost': 10000}, {'id': 'S70', 'elements': ['M'], 'cost': 10000}, {'id': 'S71', 'elements': ['N'], 'cost': 10000}, {'id': 'S72', 'elements': ['O'], 'cost': 10000}]}","['S12', 'S21', 'S28', 'S29', 'S55', 'S56']",21,json,names
SPP,SPP,"Recently the PTA noticed multiple combo meal offers and wants to pick some so each kid’s lunch requirement is satisfied by a single selected combo — no duplicates, no missing lunches. The plan is to add up the prices of whatever combos are picked to see the total cost, and the aim is to keep that total as low as possible. The exact bundle options, which kids need what, and the prices are given below.

# total_children=14
# total_bundles=63
bundle_id,bundle_price,covered_children
S1,20,9 10 11 12
S2,192,3 4 7
S3,118,6 9
S4,81,12 13 14
S5,340,2 3 4 5
S6,72,6 11 14
S7,129,1 2 3
S8,16,14
S9,12,8 10 12
S10,162,13 14
S11,232,7 9 10 11
S12,32,1 2 3 4
S13,69,8 9 11
S14,54,11 14
S15,168,12 13
S16,80,4 5
S17,252,8 10 11 12
S18,291,6 7 8
S19,18,5 6 7
S20,192,7 8 9
S21,186,10 11 12
S22,64,2 4
S23,219,7 9 10
S24,340,4 5 6 7
S25,273,10 12 13
S26,116,10 11
S27,84,4 6 7
S28,66,3 5 7
S29,234,9 10 11
S30,261,4 5 8
S31,60,8 9 10 11
S32,336,10 12 13 14
S33,110,3 6
S34,95,5
S35,78,8 9 10
S36,56,12
S37,20,11 12 13 14
S38,122,2 3
S39,180,9 12
S40,90,2 3 4
S41,276,9 10 12
S42,162,2 4 5
S43,84,10 11 12 14
S44,69,3 5 6
S45,136,5 6 7 8
S46,130,2 5
S47,15,2
S48,138,2 3 6
S49,177,9 10 13
S50,10000,1
S51,10000,2
S52,10000,3
S53,10000,4
S54,10000,5
S55,10000,6
S56,10000,7
S57,10000,8
S58,10000,9
S59,10000,10
S60,10000,11
S61,10000,12
S62,10000,13
S63,10000,14

If you want to tell me which combos the PTA will pick, a simple JSON sketch works great — something like this:

{
  ""solution"": [""combo_id""]
}

""solution"" is just the list of combo identifiers you’re choosing (one entry per selected combo). Think of it as a tiny form: drop the combo IDs in that array and that’s the set of combos you picked. This is only a sketch of the expected shape, not the actual answer.

Please make sure to use the identifiers exactly as they appear in the instance input — no renaming and no new labels.
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 14, 'num_sets': 63, 'density': 0.17346938775510204, 'sets': [{'id': 1, 'elements': [9, 10, 11, 12], 'cost': 20}, {'id': 2, 'elements': [3, 4, 7], 'cost': 192}, {'id': 3, 'elements': [6, 9], 'cost': 118}, {'id': 4, 'elements': [12, 13, 14], 'cost': 81}, {'id': 5, 'elements': [2, 3, 4, 5], 'cost': 340}, {'id': 6, 'elements': [6, 11, 14], 'cost': 72}, {'id': 7, 'elements': [1, 2, 3], 'cost': 129}, {'id': 8, 'elements': [14], 'cost': 16}, {'id': 9, 'elements': [8, 10, 12], 'cost': 12}, {'id': 10, 'elements': [13, 14], 'cost': 162}, {'id': 11, 'elements': [7, 9, 10, 11], 'cost': 232}, {'id': 12, 'elements': [1, 2, 3, 4], 'cost': 32}, {'id': 13, 'elements': [8, 9, 11], 'cost': 69}, {'id': 14, 'elements': [11, 14], 'cost': 54}, {'id': 15, 'elements': [12, 13], 'cost': 168}, {'id': 16, 'elements': [4, 5], 'cost': 80}, {'id': 17, 'elements': [8, 10, 11, 12], 'cost': 252}, {'id': 18, 'elements': [6, 7, 8], 'cost': 291}, {'id': 19, 'elements': [5, 6, 7], 'cost': 18}, {'id': 20, 'elements': [7, 8, 9], 'cost': 192}, {'id': 21, 'elements': [10, 11, 12], 'cost': 186}, {'id': 22, 'elements': [2, 4], 'cost': 64}, {'id': 23, 'elements': [7, 9, 10], 'cost': 219}, {'id': 24, 'elements': [4, 5, 6, 7], 'cost': 340}, {'id': 25, 'elements': [10, 12, 13], 'cost': 273}, {'id': 26, 'elements': [10, 11], 'cost': 116}, {'id': 27, 'elements': [4, 6, 7], 'cost': 84}, {'id': 28, 'elements': [3, 5, 7], 'cost': 66}, {'id': 29, 'elements': [9, 10, 11], 'cost': 234}, {'id': 30, 'elements': [4, 5, 8], 'cost': 261}, {'id': 31, 'elements': [8, 9, 10, 11], 'cost': 60}, {'id': 32, 'elements': [10, 12, 13, 14], 'cost': 336}, {'id': 33, 'elements': [3, 6], 'cost': 110}, {'id': 34, 'elements': [5], 'cost': 95}, {'id': 35, 'elements': [8, 9, 10], 'cost': 78}, {'id': 36, 'elements': [12], 'cost': 56}, {'id': 37, 'elements': [11, 12, 13, 14], 'cost': 20}, {'id': 38, 'elements': [2, 3], 'cost': 122}, {'id': 39, 'elements': [9, 12], 'cost': 180}, {'id': 40, 'elements': [2, 3, 4], 'cost': 90}, {'id': 41, 'elements': [9, 10, 12], 'cost': 276}, {'id': 42, 'elements': [2, 4, 5], 'cost': 162}, {'id': 43, 'elements': [10, 11, 12, 14], 'cost': 84}, {'id': 44, 'elements': [3, 5, 6], 'cost': 69}, {'id': 45, 'elements': [5, 6, 7, 8], 'cost': 136}, {'id': 46, 'elements': [2, 5], 'cost': 130}, {'id': 47, 'elements': [2], 'cost': 15}, {'id': 48, 'elements': [2, 3, 6], 'cost': 138}, {'id': 49, 'elements': [9, 10, 13], 'cost': 177}, {'id': 50, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [14], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0021_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0021_bag_stats.png'}","[12, 19, 35, 37]",148.0,"{'num_elements': 14, 'num_sets': 63, 'sets': [{'id': 'S1', 'elements': [9, 10, 11, 12], 'cost': 20}, {'id': 'S2', 'elements': [3, 4, 7], 'cost': 192}, {'id': 'S3', 'elements': [6, 9], 'cost': 118}, {'id': 'S4', 'elements': [12, 13, 14], 'cost': 81}, {'id': 'S5', 'elements': [2, 3, 4, 5], 'cost': 340}, {'id': 'S6', 'elements': [6, 11, 14], 'cost': 72}, {'id': 'S7', 'elements': [1, 2, 3], 'cost': 129}, {'id': 'S8', 'elements': [14], 'cost': 16}, {'id': 'S9', 'elements': [8, 10, 12], 'cost': 12}, {'id': 'S10', 'elements': [13, 14], 'cost': 162}, {'id': 'S11', 'elements': [7, 9, 10, 11], 'cost': 232}, {'id': 'S12', 'elements': [1, 2, 3, 4], 'cost': 32}, {'id': 'S13', 'elements': [8, 9, 11], 'cost': 69}, {'id': 'S14', 'elements': [11, 14], 'cost': 54}, {'id': 'S15', 'elements': [12, 13], 'cost': 168}, {'id': 'S16', 'elements': [4, 5], 'cost': 80}, {'id': 'S17', 'elements': [8, 10, 11, 12], 'cost': 252}, {'id': 'S18', 'elements': [6, 7, 8], 'cost': 291}, {'id': 'S19', 'elements': [5, 6, 7], 'cost': 18}, {'id': 'S20', 'elements': [7, 8, 9], 'cost': 192}, {'id': 'S21', 'elements': [10, 11, 12], 'cost': 186}, {'id': 'S22', 'elements': [2, 4], 'cost': 64}, {'id': 'S23', 'elements': [7, 9, 10], 'cost': 219}, {'id': 'S24', 'elements': [4, 5, 6, 7], 'cost': 340}, {'id': 'S25', 'elements': [10, 12, 13], 'cost': 273}, {'id': 'S26', 'elements': [10, 11], 'cost': 116}, {'id': 'S27', 'elements': [4, 6, 7], 'cost': 84}, {'id': 'S28', 'elements': [3, 5, 7], 'cost': 66}, {'id': 'S29', 'elements': [9, 10, 11], 'cost': 234}, {'id': 'S30', 'elements': [4, 5, 8], 'cost': 261}, {'id': 'S31', 'elements': [8, 9, 10, 11], 'cost': 60}, {'id': 'S32', 'elements': [10, 12, 13, 14], 'cost': 336}, {'id': 'S33', 'elements': [3, 6], 'cost': 110}, {'id': 'S34', 'elements': [5], 'cost': 95}, {'id': 'S35', 'elements': [8, 9, 10], 'cost': 78}, {'id': 'S36', 'elements': [12], 'cost': 56}, {'id': 'S37', 'elements': [11, 12, 13, 14], 'cost': 20}, {'id': 'S38', 'elements': [2, 3], 'cost': 122}, {'id': 'S39', 'elements': [9, 12], 'cost': 180}, {'id': 'S40', 'elements': [2, 3, 4], 'cost': 90}, {'id': 'S41', 'elements': [9, 10, 12], 'cost': 276}, {'id': 'S42', 'elements': [2, 4, 5], 'cost': 162}, {'id': 'S43', 'elements': [10, 11, 12, 14], 'cost': 84}, {'id': 'S44', 'elements': [3, 5, 6], 'cost': 69}, {'id': 'S45', 'elements': [5, 6, 7, 8], 'cost': 136}, {'id': 'S46', 'elements': [2, 5], 'cost': 130}, {'id': 'S47', 'elements': [2], 'cost': 15}, {'id': 'S48', 'elements': [2, 3, 6], 'cost': 138}, {'id': 'S49', 'elements': [9, 10, 13], 'cost': 177}, {'id': 'S50', 'elements': [1], 'cost': 10000}, {'id': 'S51', 'elements': [2], 'cost': 10000}, {'id': 'S52', 'elements': [3], 'cost': 10000}, {'id': 'S53', 'elements': [4], 'cost': 10000}, {'id': 'S54', 'elements': [5], 'cost': 10000}, {'id': 'S55', 'elements': [6], 'cost': 10000}, {'id': 'S56', 'elements': [7], 'cost': 10000}, {'id': 'S57', 'elements': [8], 'cost': 10000}, {'id': 'S58', 'elements': [9], 'cost': 10000}, {'id': 'S59', 'elements': [10], 'cost': 10000}, {'id': 'S60', 'elements': [11], 'cost': 10000}, {'id': 'S61', 'elements': [12], 'cost': 10000}, {'id': 'S62', 'elements': [13], 'cost': 10000}, {'id': 'S63', 'elements': [14], 'cost': 10000}]}","['S12', 'S19', 'S35', 'S37']",22,csv,1
SPP,SPP,"We’ve got a pile of bundled displays to choose from and a full catalog of product models that need to be represented. The job is to decide which bundles to stock so every single model shows up in one, and only one, of the chosen bundles — no gaps, no repeats. To judge which selection is best, total up the purchase price of the bundles you pick and go with the smallest sum. See the bundle choices and prices below.

{
  ""total_product_models"": 13,
  ""total_bundle_options"": 61,
  ""sets"": [
    {
      ""bundle_id"": ""S1"",
      ""bundle_models"": [
        ""J"",
        ""K"",
        ""L""
      ],
      ""bundle_cost"": 18
    },
    {
      ""bundle_id"": ""S2"",
      ""bundle_models"": [
        ""C"",
        ""D"",
        ""E""
      ],
      ""bundle_cost"": 63
    },
    {
      ""bundle_id"": ""S3"",
      ""bundle_models"": [
        ""K"",
        ""L"",
        ""M""
      ],
      ""bundle_cost"": 39
    },
    {
      ""bundle_id"": ""S4"",
      ""bundle_models"": [
        ""D"",
        ""E""
      ],
      ""bundle_cost"": 52
    },
    {
      ""bundle_id"": ""S5"",
      ""bundle_models"": [
        ""D"",
        ""H""
      ],
      ""bundle_cost"": 122
    },
    {
      ""bundle_id"": ""S6"",
      ""bundle_models"": [
        ""H"",
        ""I""
      ],
      ""bundle_cost"": 184
    },
    {
      ""bundle_id"": ""S7"",
      ""bundle_models"": [
        ""E"",
        ""F"",
        ""H""
      ],
      ""bundle_cost"": 204
    },
    {
      ""bundle_id"": ""S8"",
      ""bundle_models"": [
        ""C"",
        ""D""
      ],
      ""bundle_cost"": 106
    },
    {
      ""bundle_id"": ""S9"",
      ""bundle_models"": [
        ""F"",
        ""G"",
        ""H"",
        ""J""
      ],
      ""bundle_cost"": 8
    },
    {
      ""bundle_id"": ""S10"",
      ""bundle_models"": [
        ""G"",
        ""I"",
        ""K""
      ],
      ""bundle_cost"": 297
    },
    {
      ""bundle_id"": ""S11"",
      ""bundle_models"": [
        ""A"",
        ""B""
      ],
      ""bundle_cost"": 166
    },
    {
      ""bundle_id"": ""S12"",
      ""bundle_models"": [
        ""H"",
        ""K""
      ],
      ""bundle_cost"": 184
    },
    {
      ""bundle_id"": ""S13"",
      ""bundle_models"": [
        ""I"",
        ""J"",
        ""K""
      ],
      ""bundle_cost"": 270
    },
    {
      ""bundle_id"": ""S14"",
      ""bundle_models"": [
        ""D"",
        ""E"",
        ""F""
      ],
      ""bundle_cost"": 108
    },
    {
      ""bundle_id"": ""S15"",
      ""bundle_models"": [
        ""H"",
        ""I"",
        ""J"",
        ""L""
      ],
      ""bundle_cost"": 280
    },
    {
      ""bundle_id"": ""S16"",
      ""bundle_models"": [
        ""E"",
        ""H"",
        ""I""
      ],
      ""bundle_cost"": 207
    },
    {
      ""bundle_id"": ""S17"",
      ""bundle_models"": [
        ""L"",
        ""M""
      ],
      ""bundle_cost"": 134
    },
    {
      ""bundle_id"": ""S18"",
      ""bundle_models"": [
        ""C"",
        ""D"",
        ""E"",
        ""F""
      ],
      ""bundle_cost"": 328
    },
    {
      ""bundle_id"": ""S19"",
      ""bundle_models"": [
        ""A"",
        ""C""
      ],
      ""bundle_cost"": 68
    },
    {
      ""bundle_id"": ""S20"",
      ""bundle_models"": [
        ""J"",
        ""L"",
        ""M""
      ],
      ""bundle_cost"": 138
    },
    {
      ""bundle_id"": ""S21"",
      ""bundle_models"": [
        ""K"",
        ""M""
      ],
      ""bundle_cost"": 78
    },
    {
      ""bundle_id"": ""S22"",
      ""bundle_models"": [
        ""A"",
        ""C"",
        ""D""
      ],
      ""bundle_cost"": 225
    },
    {
      ""bundle_id"": ""S23"",
      ""bundle_models"": [
        ""I"",
        ""K"",
        ""L""
      ],
      ""bundle_cost"": 117
    },
    {
      ""bundle_id"": ""S24"",
      ""bundle_models"": [
        ""F"",
        ""G""
      ],
      ""bundle_cost"": 84
    },
    {
      ""bundle_id"": ""S25"",
      ""bundle_models"": [
        ""B"",
        ""C"",
        ""E""
      ],
      ""bundle_cost"": 276
    },
    {
      ""bundle_id"": ""S26"",
      ""bundle_models"": [
        ""E"",
        ""F""
      ],
      ""bundle_cost"": 34
    },
    {
      ""bundle_id"": ""S27"",
      ""bundle_models"": [
        ""F"",
        ""H"",
        ""I""
      ],
      ""bundle_cost"": 216
    },
    {
      ""bundle_id"": ""S28"",
      ""bundle_models"": [
        ""F"",
        ""G"",
        ""H""
      ],
      ""bundle_cost"": 171
    },
    {
      ""bundle_id"": ""S29"",
      ""bundle_models"": [
        ""F"",
        ""G"",
        ""I""
      ],
      ""bundle_cost"": 195
    },
    {
      ""bundle_id"": ""S30"",
      ""bundle_models"": [
        ""G"",
        ""H"",
        ""I""
      ],
      ""bundle_cost"": 288
    },
    {
      ""bundle_id"": ""S31"",
      ""bundle_models"": [
        ""I"",
        ""K""
      ],
      ""bundle_cost"": 156
    },
    {
      ""bundle_id"": ""S32"",
      ""bundle_models"": [
        ""A"",
        ""B"",
        ""C"",
        ""D""
      ],
      ""bundle_cost"": 192
    },
    {
      ""bundle_id"": ""S33"",
      ""bundle_models"": [
        ""B"",
        ""D"",
        ""E""
      ],
      ""bundle_cost"": 120
    },
    {
      ""bundle_id"": ""S34"",
      ""bundle_models"": [
        ""A"",
        ""C"",
        ""D"",
        ""E""
      ],
      ""bundle_cost"": 36
    },
    {
      ""bundle_id"": ""S35"",
      ""bundle_models"": [
        ""E"",
        ""F"",
        ""G"",
        ""H""
      ],
      ""bundle_cost"": 196
    },
    {
      ""bundle_id"": ""S36"",
      ""bundle_models"": [
        ""J"",
        ""K"",
        ""M""
      ],
      ""bundle_cost"": 246
    },
    {
      ""bundle_id"": ""S37"",
      ""bundle_models"": [
        ""F"",
        ""I""
      ],
      ""bundle_cost"": 188
    },
    {
      ""bundle_id"": ""S38"",
      ""bundle_models"": [
        ""H"",
        ""I"",
        ""J""
      ],
      ""bundle_cost"": 273
    },
    {
      ""bundle_id"": ""S39"",
      ""bundle_models"": [
        ""L""
      ],
      ""bundle_cost"": 91
    },
    {
      ""bundle_id"": ""S40"",
      ""bundle_models"": [
        ""J"",
        ""K""
      ],
      ""bundle_cost"": 50
    },
    {
      ""bundle_id"": ""S41"",
      ""bundle_models"": [
        ""G"",
        ""H"",
        ""I"",
        ""J""
      ],
      ""bundle_cost"": 244
    },
    {
      ""bundle_id"": ""S42"",
      ""bundle_models"": [
        ""A"",
        ""B"",
        ""D""
      ],
      ""bundle_cost"": 9
    },
    {
      ""bundle_id"": ""S43"",
      ""bundle_models"": [
        ""C"",
        ""E""
      ],
      ""bundle_cost"": 150
    },
    {
      ""bundle_id"": ""S44"",
      ""bundle_models"": [
        ""K"",
        ""L""
      ],
      ""bundle_cost"": 106
    },
    {
      ""bundle_id"": ""S45"",
      ""bundle_models"": [
        ""A"",
        ""B"",
        ""C""
      ],
      ""bundle_cost"": 24
    },
    {
      ""bundle_id"": ""S46"",
      ""bundle_models"": [
        ""B"",
        ""C""
      ],
      ""bundle_cost"": 84
    },
    {
      ""bundle_id"": ""S47"",
      ""bundle_models"": [
        ""A""
      ],
      ""bundle_cost"": 30
    },
    {
      ""bundle_id"": ""S48"",
      ""bundle_models"": [
        ""H"",
        ""I"",
        ""K"",
        ""L""
      ],
      ""bundle_cost"": 128
    },
    {
      ""bundle_id"": ""S49"",
      ""bundle_models"": [
        ""A""
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S50"",
      ""bundle_models"": [
        ""B""
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S51"",
      ""bundle_models"": [
        ""C""
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S52"",
      ""bundle_models"": [
        ""D""
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S53"",
      ""bundle_models"": [
        ""E""
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S54"",
      ""bundle_models"": [
        ""F""
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S55"",
      ""bundle_models"": [
        ""G""
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S56"",
      ""bundle_models"": [
        ""H""
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S57"",
      ""bundle_models"": [
        ""I""
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S58"",
      ""bundle_models"": [
        ""J""
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S59"",
      ""bundle_models"": [
        ""K""
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S60"",
      ""bundle_models"": [
        ""L""
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S61"",
      ""bundle_models"": [
        ""M""
      ],
      ""bundle_cost"": 10000
    }
  ]
}

Oh, and when you reply with your chosen bundles, please use this simple JSON layout so it's easy to parse:

{
  ""solution"": [""bundle_id"", ...]
}

""solution"" is the list of bundle IDs you want to stock, and ""bundle_id"" is just a placeholder showing the format — replace each placeholder with the exact bundle identifier from the instance. This JSON is just a sketch of the expected shape, not your actual answer.

Please make sure to use the identifiers exactly as they appear in the instance input — no renaming and no new labels.
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 13, 'num_sets': 61, 'density': 0.18158890290037832, 'sets': [{'id': 1, 'elements': [10, 11, 12], 'cost': 18}, {'id': 2, 'elements': [3, 4, 5], 'cost': 63}, {'id': 3, 'elements': [11, 12, 13], 'cost': 39}, {'id': 4, 'elements': [4, 5], 'cost': 52}, {'id': 5, 'elements': [4, 8], 'cost': 122}, {'id': 6, 'elements': [8, 9], 'cost': 184}, {'id': 7, 'elements': [5, 6, 8], 'cost': 204}, {'id': 8, 'elements': [3, 4], 'cost': 106}, {'id': 9, 'elements': [6, 7, 8, 10], 'cost': 8}, {'id': 10, 'elements': [7, 9, 11], 'cost': 297}, {'id': 11, 'elements': [1, 2], 'cost': 166}, {'id': 12, 'elements': [8, 11], 'cost': 184}, {'id': 13, 'elements': [9, 10, 11], 'cost': 270}, {'id': 14, 'elements': [4, 5, 6], 'cost': 108}, {'id': 15, 'elements': [8, 9, 10, 12], 'cost': 280}, {'id': 16, 'elements': [5, 8, 9], 'cost': 207}, {'id': 17, 'elements': [12, 13], 'cost': 134}, {'id': 18, 'elements': [3, 4, 5, 6], 'cost': 328}, {'id': 19, 'elements': [1, 3], 'cost': 68}, {'id': 20, 'elements': [10, 12, 13], 'cost': 138}, {'id': 21, 'elements': [11, 13], 'cost': 78}, {'id': 22, 'elements': [1, 3, 4], 'cost': 225}, {'id': 23, 'elements': [9, 11, 12], 'cost': 117}, {'id': 24, 'elements': [6, 7], 'cost': 84}, {'id': 25, 'elements': [2, 3, 5], 'cost': 276}, {'id': 26, 'elements': [5, 6], 'cost': 34}, {'id': 27, 'elements': [6, 8, 9], 'cost': 216}, {'id': 28, 'elements': [6, 7, 8], 'cost': 171}, {'id': 29, 'elements': [6, 7, 9], 'cost': 195}, {'id': 30, 'elements': [7, 8, 9], 'cost': 288}, {'id': 31, 'elements': [9, 11], 'cost': 156}, {'id': 32, 'elements': [1, 2, 3, 4], 'cost': 192}, {'id': 33, 'elements': [2, 4, 5], 'cost': 120}, {'id': 34, 'elements': [1, 3, 4, 5], 'cost': 36}, {'id': 35, 'elements': [5, 6, 7, 8], 'cost': 196}, {'id': 36, 'elements': [10, 11, 13], 'cost': 246}, {'id': 37, 'elements': [6, 9], 'cost': 188}, {'id': 38, 'elements': [8, 9, 10], 'cost': 273}, {'id': 39, 'elements': [12], 'cost': 91}, {'id': 40, 'elements': [10, 11], 'cost': 50}, {'id': 41, 'elements': [7, 8, 9, 10], 'cost': 244}, {'id': 42, 'elements': [1, 2, 4], 'cost': 9}, {'id': 43, 'elements': [3, 5], 'cost': 150}, {'id': 44, 'elements': [11, 12], 'cost': 106}, {'id': 45, 'elements': [1, 2, 3], 'cost': 24}, {'id': 46, 'elements': [2, 3], 'cost': 84}, {'id': 47, 'elements': [1], 'cost': 30}, {'id': 48, 'elements': [8, 9, 11, 12], 'cost': 128}, {'id': 49, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [13], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0022_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0022_bag_stats.png'}","[4, 9, 17, 31, 45]",374.0,"{'num_elements': 13, 'num_sets': 61, 'sets': [{'id': 'S1', 'elements': ['J', 'K', 'L'], 'cost': 18}, {'id': 'S2', 'elements': ['C', 'D', 'E'], 'cost': 63}, {'id': 'S3', 'elements': ['K', 'L', 'M'], 'cost': 39}, {'id': 'S4', 'elements': ['D', 'E'], 'cost': 52}, {'id': 'S5', 'elements': ['D', 'H'], 'cost': 122}, {'id': 'S6', 'elements': ['H', 'I'], 'cost': 184}, {'id': 'S7', 'elements': ['E', 'F', 'H'], 'cost': 204}, {'id': 'S8', 'elements': ['C', 'D'], 'cost': 106}, {'id': 'S9', 'elements': ['F', 'G', 'H', 'J'], 'cost': 8}, {'id': 'S10', 'elements': ['G', 'I', 'K'], 'cost': 297}, {'id': 'S11', 'elements': ['A', 'B'], 'cost': 166}, {'id': 'S12', 'elements': ['H', 'K'], 'cost': 184}, {'id': 'S13', 'elements': ['I', 'J', 'K'], 'cost': 270}, {'id': 'S14', 'elements': ['D', 'E', 'F'], 'cost': 108}, {'id': 'S15', 'elements': ['H', 'I', 'J', 'L'], 'cost': 280}, {'id': 'S16', 'elements': ['E', 'H', 'I'], 'cost': 207}, {'id': 'S17', 'elements': ['L', 'M'], 'cost': 134}, {'id': 'S18', 'elements': ['C', 'D', 'E', 'F'], 'cost': 328}, {'id': 'S19', 'elements': ['A', 'C'], 'cost': 68}, {'id': 'S20', 'elements': ['J', 'L', 'M'], 'cost': 138}, {'id': 'S21', 'elements': ['K', 'M'], 'cost': 78}, {'id': 'S22', 'elements': ['A', 'C', 'D'], 'cost': 225}, {'id': 'S23', 'elements': ['I', 'K', 'L'], 'cost': 117}, {'id': 'S24', 'elements': ['F', 'G'], 'cost': 84}, {'id': 'S25', 'elements': ['B', 'C', 'E'], 'cost': 276}, {'id': 'S26', 'elements': ['E', 'F'], 'cost': 34}, {'id': 'S27', 'elements': ['F', 'H', 'I'], 'cost': 216}, {'id': 'S28', 'elements': ['F', 'G', 'H'], 'cost': 171}, {'id': 'S29', 'elements': ['F', 'G', 'I'], 'cost': 195}, {'id': 'S30', 'elements': ['G', 'H', 'I'], 'cost': 288}, {'id': 'S31', 'elements': ['I', 'K'], 'cost': 156}, {'id': 'S32', 'elements': ['A', 'B', 'C', 'D'], 'cost': 192}, {'id': 'S33', 'elements': ['B', 'D', 'E'], 'cost': 120}, {'id': 'S34', 'elements': ['A', 'C', 'D', 'E'], 'cost': 36}, {'id': 'S35', 'elements': ['E', 'F', 'G', 'H'], 'cost': 196}, {'id': 'S36', 'elements': ['J', 'K', 'M'], 'cost': 246}, {'id': 'S37', 'elements': ['F', 'I'], 'cost': 188}, {'id': 'S38', 'elements': ['H', 'I', 'J'], 'cost': 273}, {'id': 'S39', 'elements': ['L'], 'cost': 91}, {'id': 'S40', 'elements': ['J', 'K'], 'cost': 50}, {'id': 'S41', 'elements': ['G', 'H', 'I', 'J'], 'cost': 244}, {'id': 'S42', 'elements': ['A', 'B', 'D'], 'cost': 9}, {'id': 'S43', 'elements': ['C', 'E'], 'cost': 150}, {'id': 'S44', 'elements': ['K', 'L'], 'cost': 106}, {'id': 'S45', 'elements': ['A', 'B', 'C'], 'cost': 24}, {'id': 'S46', 'elements': ['B', 'C'], 'cost': 84}, {'id': 'S47', 'elements': ['A'], 'cost': 30}, {'id': 'S48', 'elements': ['H', 'I', 'K', 'L'], 'cost': 128}, {'id': 'S49', 'elements': ['A'], 'cost': 10000}, {'id': 'S50', 'elements': ['B'], 'cost': 10000}, {'id': 'S51', 'elements': ['C'], 'cost': 10000}, {'id': 'S52', 'elements': ['D'], 'cost': 10000}, {'id': 'S53', 'elements': ['E'], 'cost': 10000}, {'id': 'S54', 'elements': ['F'], 'cost': 10000}, {'id': 'S55', 'elements': ['G'], 'cost': 10000}, {'id': 'S56', 'elements': ['H'], 'cost': 10000}, {'id': 'S57', 'elements': ['I'], 'cost': 10000}, {'id': 'S58', 'elements': ['J'], 'cost': 10000}, {'id': 'S59', 'elements': ['K'], 'cost': 10000}, {'id': 'S60', 'elements': ['L'], 'cost': 10000}, {'id': 'S61', 'elements': ['M'], 'cost': 10000}]}","['S4', 'S9', 'S17', 'S31', 'S45']",23,json,names
SPP,SPP,"We have a bunch of students and a bunch of ready-made project kits to choose from; the task is to decide which kits to buy so each student is covered by exactly a single kit. The way to judge any choice is by totaling the prices of the kits purchased — the lower that sum, the better the plan. It’s important that no student is skipped and no student gets more than one kit, and the concrete list of students, kits, and costs appears below.

- **num_students**: 17
- **num_kits**: 80

| kit_id | kit_cost | covered_students |
|---|---|---|
| S1 | 110 | L N O P Q |
| S2 | 176 | N O |
| S3 | 111 | K M N |
| S4 | 32 | B C |
| S5 | 220 | H I J K L |
| S6 | 308 | M N O P |
| S7 | 6 | A E F |
| S8 | 44 | L M N O |
| S9 | 64 | B C D E |
| S10 | 297 | K L O |
| S11 | 384 | A B C D |
| S12 | 50 | I J |
| S13 | 128 | H K |
| S14 | 410 | A B C D E |
| S15 | 255 | H I J |
| S16 | 292 | C D E F |
| S17 | 348 | D H J K |
| S18 | 87 | O P Q |
| S19 | 102 | M N |
| S20 | 190 | D E |
| S21 | 52 | P Q |
| S22 | 8 | G K |
| S23 | 99 | A B C |
| S24 | 129 | M O P |
| S25 | 294 | H I K |
| S26 | 16 | L O |
| S27 | 285 | J K L O P |
| S28 | 144 | K O |
| S29 | 172 | C D E G |
| S30 | 100 | A B E F |
| S31 | 160 | C D E F G |
| S32 | 112 | A B |
| S33 | 35 | M N O P Q |
| S34 | 27 | E G J |
| S35 | 177 | I J K |
| S36 | 141 | K L Q |
| S37 | 12 | F H I |
| S38 | 186 | H L |
| S39 | 68 | E F |
| S40 | 170 | N P |
| S41 | 316 | H I K O |
| S42 | 68 | G I K L |
| S43 | 330 | L M N P Q |
| S44 | 276 | N P Q |
| S45 | 156 | N O P |
| S46 | 112 | A B C E |
| S47 | 213 | L M P |
| S48 | 60 | B D |
| S49 | 130 | A C |
| S50 | 30 | A D F |
| S51 | 288 | E F H |
| S52 | 368 | G H J K |
| S53 | 208 | E H J K |
| S54 | 142 | G H |
| S55 | 360 | L N O Q |
| S56 | 184 | D E F G |
| S57 | 80 | G I L M |
| S58 | 245 | I J K M N |
| S59 | 122 | J M |
| S60 | 126 | C G |
| S61 | 74 | E J |
| S62 | 106 | M P |
| S63 | 150 | B D F |
| S64 | 10000 | A |
| S65 | 10000 | B |
| S66 | 10000 | C |
| S67 | 10000 | D |
| S68 | 10000 | E |
| S69 | 10000 | F |
| S70 | 10000 | G |
| S71 | 10000 | H |
| S72 | 10000 | I |
| S73 | 10000 | J |
| S74 | 10000 | K |
| S75 | 10000 | L |
| S76 | 10000 | M |
| S77 | 10000 | N |
| S78 | 10000 | O |
| S79 | 10000 | P |
| S80 | 10000 | Q |

Also, when you send back your pick, tuck it into a tiny JSON snippet so it's easy to parse. Something like this will do:

{
  ""solution"": [""kit_id"", ...]
}

In plain words: ""solution"" should list the kits you want to buy, and each item in that array is the kit's identifier (the ""kit_id"" placeholder is just there to show the shape — replace it with the real ids from the instance). This is just a sketch of the expected shape, not the actual answer.

Please make sure you use the identifiers exactly as they're given in the instance input — don't rename them or invent new labels.  
Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'problem_type': 'SPP', 'num_elements': 17, 'num_sets': 80, 'density': 0.15955882352941175, 'sets': [{'id': 1, 'elements': [12, 14, 15, 16, 17], 'cost': 110}, {'id': 2, 'elements': [14, 15], 'cost': 176}, {'id': 3, 'elements': [11, 13, 14], 'cost': 111}, {'id': 4, 'elements': [2, 3], 'cost': 32}, {'id': 5, 'elements': [8, 9, 10, 11, 12], 'cost': 220}, {'id': 6, 'elements': [13, 14, 15, 16], 'cost': 308}, {'id': 7, 'elements': [1, 5, 6], 'cost': 6}, {'id': 8, 'elements': [12, 13, 14, 15], 'cost': 44}, {'id': 9, 'elements': [2, 3, 4, 5], 'cost': 64}, {'id': 10, 'elements': [11, 12, 15], 'cost': 297}, {'id': 11, 'elements': [1, 2, 3, 4], 'cost': 384}, {'id': 12, 'elements': [9, 10], 'cost': 50}, {'id': 13, 'elements': [8, 11], 'cost': 128}, {'id': 14, 'elements': [1, 2, 3, 4, 5], 'cost': 410}, {'id': 15, 'elements': [8, 9, 10], 'cost': 255}, {'id': 16, 'elements': [3, 4, 5, 6], 'cost': 292}, {'id': 17, 'elements': [4, 8, 10, 11], 'cost': 348}, {'id': 18, 'elements': [15, 16, 17], 'cost': 87}, {'id': 19, 'elements': [13, 14], 'cost': 102}, {'id': 20, 'elements': [4, 5], 'cost': 190}, {'id': 21, 'elements': [16, 17], 'cost': 52}, {'id': 22, 'elements': [7, 11], 'cost': 8}, {'id': 23, 'elements': [1, 2, 3], 'cost': 99}, {'id': 24, 'elements': [13, 15, 16], 'cost': 129}, {'id': 25, 'elements': [8, 9, 11], 'cost': 294}, {'id': 26, 'elements': [12, 15], 'cost': 16}, {'id': 27, 'elements': [10, 11, 12, 15, 16], 'cost': 285}, {'id': 28, 'elements': [11, 15], 'cost': 144}, {'id': 29, 'elements': [3, 4, 5, 7], 'cost': 172}, {'id': 30, 'elements': [1, 2, 5, 6], 'cost': 100}, {'id': 31, 'elements': [3, 4, 5, 6, 7], 'cost': 160}, {'id': 32, 'elements': [1, 2], 'cost': 112}, {'id': 33, 'elements': [13, 14, 15, 16, 17], 'cost': 35}, {'id': 34, 'elements': [5, 7, 10], 'cost': 27}, {'id': 35, 'elements': [9, 10, 11], 'cost': 177}, {'id': 36, 'elements': [11, 12, 17], 'cost': 141}, {'id': 37, 'elements': [6, 8, 9], 'cost': 12}, {'id': 38, 'elements': [8, 12], 'cost': 186}, {'id': 39, 'elements': [5, 6], 'cost': 68}, {'id': 40, 'elements': [14, 16], 'cost': 170}, {'id': 41, 'elements': [8, 9, 11, 15], 'cost': 316}, {'id': 42, 'elements': [7, 9, 11, 12], 'cost': 68}, {'id': 43, 'elements': [12, 13, 14, 16, 17], 'cost': 330}, {'id': 44, 'elements': [14, 16, 17], 'cost': 276}, {'id': 45, 'elements': [14, 15, 16], 'cost': 156}, {'id': 46, 'elements': [1, 2, 3, 5], 'cost': 112}, {'id': 47, 'elements': [12, 13, 16], 'cost': 213}, {'id': 48, 'elements': [2, 4], 'cost': 60}, {'id': 49, 'elements': [1, 3], 'cost': 130}, {'id': 50, 'elements': [1, 4, 6], 'cost': 30}, {'id': 51, 'elements': [5, 6, 8], 'cost': 288}, {'id': 52, 'elements': [7, 8, 10, 11], 'cost': 368}, {'id': 53, 'elements': [5, 8, 10, 11], 'cost': 208}, {'id': 54, 'elements': [7, 8], 'cost': 142}, {'id': 55, 'elements': [12, 14, 15, 17], 'cost': 360}, {'id': 56, 'elements': [4, 5, 6, 7], 'cost': 184}, {'id': 57, 'elements': [7, 9, 12, 13], 'cost': 80}, {'id': 58, 'elements': [9, 10, 11, 13, 14], 'cost': 245}, {'id': 59, 'elements': [10, 13], 'cost': 122}, {'id': 60, 'elements': [3, 7], 'cost': 126}, {'id': 61, 'elements': [5, 10], 'cost': 74}, {'id': 62, 'elements': [13, 16], 'cost': 106}, {'id': 63, 'elements': [2, 4, 6], 'cost': 150}, {'id': 64, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 66, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 67, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 68, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 69, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 70, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 71, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 72, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 73, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 74, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 75, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 76, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 77, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 78, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 79, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 80, 'elements': [17], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0023_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0023_bag_stats.png'}","[8, 21, 22, 37, 48, 49, 61]",380.0,"{'num_elements': 17, 'num_sets': 80, 'sets': [{'id': 'S1', 'elements': ['L', 'N', 'O', 'P', 'Q'], 'cost': 110}, {'id': 'S2', 'elements': ['N', 'O'], 'cost': 176}, {'id': 'S3', 'elements': ['K', 'M', 'N'], 'cost': 111}, {'id': 'S4', 'elements': ['B', 'C'], 'cost': 32}, {'id': 'S5', 'elements': ['H', 'I', 'J', 'K', 'L'], 'cost': 220}, {'id': 'S6', 'elements': ['M', 'N', 'O', 'P'], 'cost': 308}, {'id': 'S7', 'elements': ['A', 'E', 'F'], 'cost': 6}, {'id': 'S8', 'elements': ['L', 'M', 'N', 'O'], 'cost': 44}, {'id': 'S9', 'elements': ['B', 'C', 'D', 'E'], 'cost': 64}, {'id': 'S10', 'elements': ['K', 'L', 'O'], 'cost': 297}, {'id': 'S11', 'elements': ['A', 'B', 'C', 'D'], 'cost': 384}, {'id': 'S12', 'elements': ['I', 'J'], 'cost': 50}, {'id': 'S13', 'elements': ['H', 'K'], 'cost': 128}, {'id': 'S14', 'elements': ['A', 'B', 'C', 'D', 'E'], 'cost': 410}, {'id': 'S15', 'elements': ['H', 'I', 'J'], 'cost': 255}, {'id': 'S16', 'elements': ['C', 'D', 'E', 'F'], 'cost': 292}, {'id': 'S17', 'elements': ['D', 'H', 'J', 'K'], 'cost': 348}, {'id': 'S18', 'elements': ['O', 'P', 'Q'], 'cost': 87}, {'id': 'S19', 'elements': ['M', 'N'], 'cost': 102}, {'id': 'S20', 'elements': ['D', 'E'], 'cost': 190}, {'id': 'S21', 'elements': ['P', 'Q'], 'cost': 52}, {'id': 'S22', 'elements': ['G', 'K'], 'cost': 8}, {'id': 'S23', 'elements': ['A', 'B', 'C'], 'cost': 99}, {'id': 'S24', 'elements': ['M', 'O', 'P'], 'cost': 129}, {'id': 'S25', 'elements': ['H', 'I', 'K'], 'cost': 294}, {'id': 'S26', 'elements': ['L', 'O'], 'cost': 16}, {'id': 'S27', 'elements': ['J', 'K', 'L', 'O', 'P'], 'cost': 285}, {'id': 'S28', 'elements': ['K', 'O'], 'cost': 144}, {'id': 'S29', 'elements': ['C', 'D', 'E', 'G'], 'cost': 172}, {'id': 'S30', 'elements': ['A', 'B', 'E', 'F'], 'cost': 100}, {'id': 'S31', 'elements': ['C', 'D', 'E', 'F', 'G'], 'cost': 160}, {'id': 'S32', 'elements': ['A', 'B'], 'cost': 112}, {'id': 'S33', 'elements': ['M', 'N', 'O', 'P', 'Q'], 'cost': 35}, {'id': 'S34', 'elements': ['E', 'G', 'J'], 'cost': 27}, {'id': 'S35', 'elements': ['I', 'J', 'K'], 'cost': 177}, {'id': 'S36', 'elements': ['K', 'L', 'Q'], 'cost': 141}, {'id': 'S37', 'elements': ['F', 'H', 'I'], 'cost': 12}, {'id': 'S38', 'elements': ['H', 'L'], 'cost': 186}, {'id': 'S39', 'elements': ['E', 'F'], 'cost': 68}, {'id': 'S40', 'elements': ['N', 'P'], 'cost': 170}, {'id': 'S41', 'elements': ['H', 'I', 'K', 'O'], 'cost': 316}, {'id': 'S42', 'elements': ['G', 'I', 'K', 'L'], 'cost': 68}, {'id': 'S43', 'elements': ['L', 'M', 'N', 'P', 'Q'], 'cost': 330}, {'id': 'S44', 'elements': ['N', 'P', 'Q'], 'cost': 276}, {'id': 'S45', 'elements': ['N', 'O', 'P'], 'cost': 156}, {'id': 'S46', 'elements': ['A', 'B', 'C', 'E'], 'cost': 112}, {'id': 'S47', 'elements': ['L', 'M', 'P'], 'cost': 213}, {'id': 'S48', 'elements': ['B', 'D'], 'cost': 60}, {'id': 'S49', 'elements': ['A', 'C'], 'cost': 130}, {'id': 'S50', 'elements': ['A', 'D', 'F'], 'cost': 30}, {'id': 'S51', 'elements': ['E', 'F', 'H'], 'cost': 288}, {'id': 'S52', 'elements': ['G', 'H', 'J', 'K'], 'cost': 368}, {'id': 'S53', 'elements': ['E', 'H', 'J', 'K'], 'cost': 208}, {'id': 'S54', 'elements': ['G', 'H'], 'cost': 142}, {'id': 'S55', 'elements': ['L', 'N', 'O', 'Q'], 'cost': 360}, {'id': 'S56', 'elements': ['D', 'E', 'F', 'G'], 'cost': 184}, {'id': 'S57', 'elements': ['G', 'I', 'L', 'M'], 'cost': 80}, {'id': 'S58', 'elements': ['I', 'J', 'K', 'M', 'N'], 'cost': 245}, {'id': 'S59', 'elements': ['J', 'M'], 'cost': 122}, {'id': 'S60', 'elements': ['C', 'G'], 'cost': 126}, {'id': 'S61', 'elements': ['E', 'J'], 'cost': 74}, {'id': 'S62', 'elements': ['M', 'P'], 'cost': 106}, {'id': 'S63', 'elements': ['B', 'D', 'F'], 'cost': 150}, {'id': 'S64', 'elements': ['A'], 'cost': 10000}, {'id': 'S65', 'elements': ['B'], 'cost': 10000}, {'id': 'S66', 'elements': ['C'], 'cost': 10000}, {'id': 'S67', 'elements': ['D'], 'cost': 10000}, {'id': 'S68', 'elements': ['E'], 'cost': 10000}, {'id': 'S69', 'elements': ['F'], 'cost': 10000}, {'id': 'S70', 'elements': ['G'], 'cost': 10000}, {'id': 'S71', 'elements': ['H'], 'cost': 10000}, {'id': 'S72', 'elements': ['I'], 'cost': 10000}, {'id': 'S73', 'elements': ['J'], 'cost': 10000}, {'id': 'S74', 'elements': ['K'], 'cost': 10000}, {'id': 'S75', 'elements': ['L'], 'cost': 10000}, {'id': 'S76', 'elements': ['M'], 'cost': 10000}, {'id': 'S77', 'elements': ['N'], 'cost': 10000}, {'id': 'S78', 'elements': ['O'], 'cost': 10000}, {'id': 'S79', 'elements': ['P'], 'cost': 10000}, {'id': 'S80', 'elements': ['Q'], 'cost': 10000}]}","['S8', 'S21', 'S22', 'S37', 'S48', 'S49', 'S61']",24,markdown_table,names
SPP,SPP,"At a local bookstore they’re prepping themed bundles and need to choose a selection of boxes so that every title is covered exactly once — every book must be in one box and cannot appear in more than one. To judge a selection, add up the cost of each chosen box; the selection with the smallest total cost is preferred. The exact titles and box choices are posted below.

{
  ""total_titles"": 14,
  ""num_boxes"": 52,
  ""sets"": [
    {
      ""box_id"": ""S1"",
      ""titles_in_box"": [
        ""G"",
        ""H""
      ],
      ""box_cost"": 42
    },
    {
      ""box_id"": ""S2"",
      ""titles_in_box"": [
        ""D"",
        ""E"",
        ""F""
      ],
      ""box_cost"": 108
    },
    {
      ""box_id"": ""S3"",
      ""titles_in_box"": [
        ""C"",
        ""E""
      ],
      ""box_cost"": 156
    },
    {
      ""box_id"": ""S4"",
      ""titles_in_box"": [
        ""G"",
        ""H"",
        ""I"",
        ""J""
      ],
      ""box_cost"": 24
    },
    {
      ""box_id"": ""S5"",
      ""titles_in_box"": [
        ""D"",
        ""F"",
        ""G""
      ],
      ""box_cost"": 276
    },
    {
      ""box_id"": ""S6"",
      ""titles_in_box"": [
        ""E"",
        ""F"",
        ""G"",
        ""I""
      ],
      ""box_cost"": 320
    },
    {
      ""box_id"": ""S7"",
      ""titles_in_box"": [
        ""G"",
        ""H"",
        ""I""
      ],
      ""box_cost"": 39
    },
    {
      ""box_id"": ""S8"",
      ""titles_in_box"": [
        ""A"",
        ""B"",
        ""C""
      ],
      ""box_cost"": 243
    },
    {
      ""box_id"": ""S9"",
      ""titles_in_box"": [
        ""K"",
        ""L"",
        ""M""
      ],
      ""box_cost"": 84
    },
    {
      ""box_id"": ""S10"",
      ""titles_in_box"": [
        ""E"",
        ""G"",
        ""J"",
        ""K""
      ],
      ""box_cost"": 56
    },
    {
      ""box_id"": ""S11"",
      ""titles_in_box"": [
        ""E"",
        ""H"",
        ""I""
      ],
      ""box_cost"": 180
    },
    {
      ""box_id"": ""S12"",
      ""titles_in_box"": [
        ""J"",
        ""L""
      ],
      ""box_cost"": 10
    },
    {
      ""box_id"": ""S13"",
      ""titles_in_box"": [
        ""C""
      ],
      ""box_cost"": 18
    },
    {
      ""box_id"": ""S14"",
      ""titles_in_box"": [
        ""K"",
        ""L""
      ],
      ""box_cost"": 22
    },
    {
      ""box_id"": ""S15"",
      ""titles_in_box"": [
        ""C"",
        ""D"",
        ""E""
      ],
      ""box_cost"": 108
    },
    {
      ""box_id"": ""S16"",
      ""titles_in_box"": [
        ""A"",
        ""B"",
        ""C"",
        ""D""
      ],
      ""box_cost"": 380
    },
    {
      ""box_id"": ""S17"",
      ""titles_in_box"": [
        ""A"",
        ""C""
      ],
      ""box_cost"": 70
    },
    {
      ""box_id"": ""S18"",
      ""titles_in_box"": [
        ""E"",
        ""G""
      ],
      ""box_cost"": 174
    },
    {
      ""box_id"": ""S19"",
      ""titles_in_box"": [
        ""H"",
        ""I"",
        ""J""
      ],
      ""box_cost"": 249
    },
    {
      ""box_id"": ""S20"",
      ""titles_in_box"": [
        ""J"",
        ""K"",
        ""M""
      ],
      ""box_cost"": 21
    },
    {
      ""box_id"": ""S21"",
      ""titles_in_box"": [
        ""B"",
        ""C"",
        ""D""
      ],
      ""box_cost"": 144
    },
    {
      ""box_id"": ""S22"",
      ""titles_in_box"": [
        ""B"",
        ""C"",
        ""D"",
        ""G""
      ],
      ""box_cost"": 100
    },
    {
      ""box_id"": ""S23"",
      ""titles_in_box"": [
        ""A"",
        ""C"",
        ""D""
      ],
      ""box_cost"": 258
    },
    {
      ""box_id"": ""S24"",
      ""titles_in_box"": [
        ""D"",
        ""E"",
        ""F"",
        ""G""
      ],
      ""box_cost"": 220
    },
    {
      ""box_id"": ""S25"",
      ""titles_in_box"": [
        ""B"",
        ""D""
      ],
      ""box_cost"": 4
    },
    {
      ""box_id"": ""S26"",
      ""titles_in_box"": [
        ""C"",
        ""E"",
        ""F""
      ],
      ""box_cost"": 255
    },
    {
      ""box_id"": ""S27"",
      ""titles_in_box"": [
        ""H"",
        ""J"",
        ""K"",
        ""M""
      ],
      ""box_cost"": 396
    },
    {
      ""box_id"": ""S28"",
      ""titles_in_box"": [
        ""E"",
        ""G"",
        ""H""
      ],
      ""box_cost"": 270
    },
    {
      ""box_id"": ""S29"",
      ""titles_in_box"": [
        ""L"",
        ""M""
      ],
      ""box_cost"": 164
    },
    {
      ""box_id"": ""S30"",
      ""titles_in_box"": [
        ""J"",
        ""K"",
        ""L"",
        ""M""
      ],
      ""box_cost"": 312
    },
    {
      ""box_id"": ""S31"",
      ""titles_in_box"": [
        ""F"",
        ""G"",
        ""H"",
        ""I""
      ],
      ""box_cost"": 8
    },
    {
      ""box_id"": ""S32"",
      ""titles_in_box"": [
        ""H"",
        ""K""
      ],
      ""box_cost"": 62
    },
    {
      ""box_id"": ""S33"",
      ""titles_in_box"": [
        ""H"",
        ""J"",
        ""L"",
        ""M""
      ],
      ""box_cost"": 164
    },
    {
      ""box_id"": ""S34"",
      ""titles_in_box"": [
        ""K"",
        ""L"",
        ""N""
      ],
      ""box_cost"": 111
    },
    {
      ""box_id"": ""S35"",
      ""titles_in_box"": [
        ""C"",
        ""D"",
        ""E"",
        ""F""
      ],
      ""box_cost"": 344
    },
    {
      ""box_id"": ""S36"",
      ""titles_in_box"": [
        ""K"",
        ""L"",
        ""M"",
        ""N""
      ],
      ""box_cost"": 168
    },
    {
      ""box_id"": ""S37"",
      ""titles_in_box"": [
        ""H"",
        ""M""
      ],
      ""box_cost"": 18
    },
    {
      ""box_id"": ""S38"",
      ""titles_in_box"": [
        ""L"",
        ""M"",
        ""N""
      ],
      ""box_cost"": 165
    },
    {
      ""box_id"": ""S39"",
      ""titles_in_box"": [
        ""A""
      ],
      ""box_cost"": 10000
    },
    {
      ""box_id"": ""S40"",
      ""titles_in_box"": [
        ""B""
      ],
      ""box_cost"": 10000
    },
    {
      ""box_id"": ""S41"",
      ""titles_in_box"": [
        ""C""
      ],
      ""box_cost"": 10000
    },
    {
      ""box_id"": ""S42"",
      ""titles_in_box"": [
        ""D""
      ],
      ""box_cost"": 10000
    },
    {
      ""box_id"": ""S43"",
      ""titles_in_box"": [
        ""E""
      ],
      ""box_cost"": 10000
    },
    {
      ""box_id"": ""S44"",
      ""titles_in_box"": [
        ""F""
      ],
      ""box_cost"": 10000
    },
    {
      ""box_id"": ""S45"",
      ""titles_in_box"": [
        ""G""
      ],
      ""box_cost"": 10000
    },
    {
      ""box_id"": ""S46"",
      ""titles_in_box"": [
        ""H""
      ],
      ""box_cost"": 10000
    },
    {
      ""box_id"": ""S47"",
      ""titles_in_box"": [
        ""I""
      ],
      ""box_cost"": 10000
    },
    {
      ""box_id"": ""S48"",
      ""titles_in_box"": [
        ""J""
      ],
      ""box_cost"": 10000
    },
    {
      ""box_id"": ""S49"",
      ""titles_in_box"": [
        ""K""
      ],
      ""box_cost"": 10000
    },
    {
      ""box_id"": ""S50"",
      ""titles_in_box"": [
        ""L""
      ],
      ""box_cost"": 10000
    },
    {
      ""box_id"": ""S51"",
      ""titles_in_box"": [
        ""M""
      ],
      ""box_cost"": 10000
    },
    {
      ""box_id"": ""S52"",
      ""titles_in_box"": [
        ""N""
      ],
      ""box_cost"": 10000
    }
  ]
}

Also, when you're ready to tell me which boxes you picked, please send it in this little JSON shape so I can read it easily:

{
  ""solution"": [""box_id"", ...]
}

Here ""solution"" is the list where you put the identifiers of the boxes you chose — each box goes in the array. The ""box_id"" is just a placeholder showing the kind of thing to put there (it's not the real answer).

This is only a sketch of the expected shape, not the actual answer. Please make sure every identifier you include matches the instance input exactly — no renaming and no new labels. 

- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 14, 'num_sets': 52, 'density': 0.17582417582417584, 'sets': [{'id': 1, 'elements': [7, 8], 'cost': 42}, {'id': 2, 'elements': [4, 5, 6], 'cost': 108}, {'id': 3, 'elements': [3, 5], 'cost': 156}, {'id': 4, 'elements': [7, 8, 9, 10], 'cost': 24}, {'id': 5, 'elements': [4, 6, 7], 'cost': 276}, {'id': 6, 'elements': [5, 6, 7, 9], 'cost': 320}, {'id': 7, 'elements': [7, 8, 9], 'cost': 39}, {'id': 8, 'elements': [1, 2, 3], 'cost': 243}, {'id': 9, 'elements': [11, 12, 13], 'cost': 84}, {'id': 10, 'elements': [5, 7, 10, 11], 'cost': 56}, {'id': 11, 'elements': [5, 8, 9], 'cost': 180}, {'id': 12, 'elements': [10, 12], 'cost': 10}, {'id': 13, 'elements': [3], 'cost': 18}, {'id': 14, 'elements': [11, 12], 'cost': 22}, {'id': 15, 'elements': [3, 4, 5], 'cost': 108}, {'id': 16, 'elements': [1, 2, 3, 4], 'cost': 380}, {'id': 17, 'elements': [1, 3], 'cost': 70}, {'id': 18, 'elements': [5, 7], 'cost': 174}, {'id': 19, 'elements': [8, 9, 10], 'cost': 249}, {'id': 20, 'elements': [10, 11, 13], 'cost': 21}, {'id': 21, 'elements': [2, 3, 4], 'cost': 144}, {'id': 22, 'elements': [2, 3, 4, 7], 'cost': 100}, {'id': 23, 'elements': [1, 3, 4], 'cost': 258}, {'id': 24, 'elements': [4, 5, 6, 7], 'cost': 220}, {'id': 25, 'elements': [2, 4], 'cost': 4}, {'id': 26, 'elements': [3, 5, 6], 'cost': 255}, {'id': 27, 'elements': [8, 10, 11, 13], 'cost': 396}, {'id': 28, 'elements': [5, 7, 8], 'cost': 270}, {'id': 29, 'elements': [12, 13], 'cost': 164}, {'id': 30, 'elements': [10, 11, 12, 13], 'cost': 312}, {'id': 31, 'elements': [6, 7, 8, 9], 'cost': 8}, {'id': 32, 'elements': [8, 11], 'cost': 62}, {'id': 33, 'elements': [8, 10, 12, 13], 'cost': 164}, {'id': 34, 'elements': [11, 12, 14], 'cost': 111}, {'id': 35, 'elements': [3, 4, 5, 6], 'cost': 344}, {'id': 36, 'elements': [11, 12, 13, 14], 'cost': 168}, {'id': 37, 'elements': [8, 13], 'cost': 18}, {'id': 38, 'elements': [12, 13, 14], 'cost': 165}, {'id': 39, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 40, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 41, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 42, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 43, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 44, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 45, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 46, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 47, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [14], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0024_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0024_bag_stats.png'}","[2, 4, 8, 36]",543.0,"{'num_elements': 14, 'num_sets': 52, 'sets': [{'id': 'S1', 'elements': ['G', 'H'], 'cost': 42}, {'id': 'S2', 'elements': ['D', 'E', 'F'], 'cost': 108}, {'id': 'S3', 'elements': ['C', 'E'], 'cost': 156}, {'id': 'S4', 'elements': ['G', 'H', 'I', 'J'], 'cost': 24}, {'id': 'S5', 'elements': ['D', 'F', 'G'], 'cost': 276}, {'id': 'S6', 'elements': ['E', 'F', 'G', 'I'], 'cost': 320}, {'id': 'S7', 'elements': ['G', 'H', 'I'], 'cost': 39}, {'id': 'S8', 'elements': ['A', 'B', 'C'], 'cost': 243}, {'id': 'S9', 'elements': ['K', 'L', 'M'], 'cost': 84}, {'id': 'S10', 'elements': ['E', 'G', 'J', 'K'], 'cost': 56}, {'id': 'S11', 'elements': ['E', 'H', 'I'], 'cost': 180}, {'id': 'S12', 'elements': ['J', 'L'], 'cost': 10}, {'id': 'S13', 'elements': ['C'], 'cost': 18}, {'id': 'S14', 'elements': ['K', 'L'], 'cost': 22}, {'id': 'S15', 'elements': ['C', 'D', 'E'], 'cost': 108}, {'id': 'S16', 'elements': ['A', 'B', 'C', 'D'], 'cost': 380}, {'id': 'S17', 'elements': ['A', 'C'], 'cost': 70}, {'id': 'S18', 'elements': ['E', 'G'], 'cost': 174}, {'id': 'S19', 'elements': ['H', 'I', 'J'], 'cost': 249}, {'id': 'S20', 'elements': ['J', 'K', 'M'], 'cost': 21}, {'id': 'S21', 'elements': ['B', 'C', 'D'], 'cost': 144}, {'id': 'S22', 'elements': ['B', 'C', 'D', 'G'], 'cost': 100}, {'id': 'S23', 'elements': ['A', 'C', 'D'], 'cost': 258}, {'id': 'S24', 'elements': ['D', 'E', 'F', 'G'], 'cost': 220}, {'id': 'S25', 'elements': ['B', 'D'], 'cost': 4}, {'id': 'S26', 'elements': ['C', 'E', 'F'], 'cost': 255}, {'id': 'S27', 'elements': ['H', 'J', 'K', 'M'], 'cost': 396}, {'id': 'S28', 'elements': ['E', 'G', 'H'], 'cost': 270}, {'id': 'S29', 'elements': ['L', 'M'], 'cost': 164}, {'id': 'S30', 'elements': ['J', 'K', 'L', 'M'], 'cost': 312}, {'id': 'S31', 'elements': ['F', 'G', 'H', 'I'], 'cost': 8}, {'id': 'S32', 'elements': ['H', 'K'], 'cost': 62}, {'id': 'S33', 'elements': ['H', 'J', 'L', 'M'], 'cost': 164}, {'id': 'S34', 'elements': ['K', 'L', 'N'], 'cost': 111}, {'id': 'S35', 'elements': ['C', 'D', 'E', 'F'], 'cost': 344}, {'id': 'S36', 'elements': ['K', 'L', 'M', 'N'], 'cost': 168}, {'id': 'S37', 'elements': ['H', 'M'], 'cost': 18}, {'id': 'S38', 'elements': ['L', 'M', 'N'], 'cost': 165}, {'id': 'S39', 'elements': ['A'], 'cost': 10000}, {'id': 'S40', 'elements': ['B'], 'cost': 10000}, {'id': 'S41', 'elements': ['C'], 'cost': 10000}, {'id': 'S42', 'elements': ['D'], 'cost': 10000}, {'id': 'S43', 'elements': ['E'], 'cost': 10000}, {'id': 'S44', 'elements': ['F'], 'cost': 10000}, {'id': 'S45', 'elements': ['G'], 'cost': 10000}, {'id': 'S46', 'elements': ['H'], 'cost': 10000}, {'id': 'S47', 'elements': ['I'], 'cost': 10000}, {'id': 'S48', 'elements': ['J'], 'cost': 10000}, {'id': 'S49', 'elements': ['K'], 'cost': 10000}, {'id': 'S50', 'elements': ['L'], 'cost': 10000}, {'id': 'S51', 'elements': ['M'], 'cost': 10000}, {'id': 'S52', 'elements': ['N'], 'cost': 10000}]}","['S2', 'S4', 'S8', 'S36']",25,json,names
SPP,SPP,"Recently the company reviewed its software needs and a stack of licensing proposals arrived, each proposal bundling different capabilities and a price. The challenge is to accept a subset of proposals so that every capability the company needs is covered by exactly one accepted proposal—no duplicates or misses—and then compute the total licensing bill by summing the prices of the accepted proposals. The cheapest total that still covers everything is the aim. The detailed proposals and numbers follow below.

# num_required_capabilities=15
# num_proposals=68
proposal_id,license_fee,capabilities_covered
S1,195,4 6 7
S2,9,7 8 9
S3,80,6 7 8 9
S4,147,2 3 4
S5,60,6 7 9
S6,216,1 2 3 4
S7,212,11 12 13 14
S8,132,12 13 14 15
S9,188,9 10 11 12
S10,234,8 9 12
S11,255,9 10 12
S12,220,11 13 14 15
S13,98,5 6
S14,36,12 13 14
S15,38,9 11
S16,80,3 6
S17,18,1 4
S18,192,5 7 8 9
S19,30,2 3
S20,294,12 13 15
S21,160,14 15
S22,98,10 11
S23,3,7 10 12
S24,3,11 13 15
S25,183,10 11 12
S26,124,9 10 12 15
S27,72,1 2 3 5
S28,126,10 13 14
S29,190,8 9
S30,190,7 8
S31,183,2 4 5
S32,142,11 14
S33,124,6 7 8 10
S34,24,3 4
S35,84,9 14
S36,356,8 9 11 12
S37,279,10 12 13
S38,90,10 12 14
S39,88,1 2
S40,16,3 7
S41,141,3 6 7
S42,114,8 10 11
S43,360,6 7 9 10
S44,32,5 10
S45,56,13 14
S46,60,10 13 15
S47,400,3 4 5 6
S48,170,7 9
S49,180,2 3 4 7
S50,360,5 6 8 10
S51,372,1 3 5 6
S52,183,11 12 14
S53,15,6 7 10
S54,10000,1
S55,10000,2
S56,10000,3
S57,10000,4
S58,10000,5
S59,10000,6
S60,10000,7
S61,10000,8
S62,10000,9
S63,10000,10
S64,10000,11
S65,10000,12
S66,10000,13
S67,10000,14
S68,10000,15

When you send the chosen bundle list back, please use this simple JSON layout so it's easy to read and add up the bill:

{
  ""solution"": [""proposal_id"", ...]
}

""solution"" should be an array of the proposals you decide to accept; each entry is the id of a chosen proposal (the placeholder ""proposal_id"" just stands in for whatever id the instance uses). This is just a sketch of the expected shape — not the actual answer.

Please make sure to use the exact identifiers from the instance input — no renaming and no invented labels.

- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 15, 'num_sets': 68, 'density': 0.1696078431372549, 'sets': [{'id': 1, 'elements': [4, 6, 7], 'cost': 195}, {'id': 2, 'elements': [7, 8, 9], 'cost': 9}, {'id': 3, 'elements': [6, 7, 8, 9], 'cost': 80}, {'id': 4, 'elements': [2, 3, 4], 'cost': 147}, {'id': 5, 'elements': [6, 7, 9], 'cost': 60}, {'id': 6, 'elements': [1, 2, 3, 4], 'cost': 216}, {'id': 7, 'elements': [11, 12, 13, 14], 'cost': 212}, {'id': 8, 'elements': [12, 13, 14, 15], 'cost': 132}, {'id': 9, 'elements': [9, 10, 11, 12], 'cost': 188}, {'id': 10, 'elements': [8, 9, 12], 'cost': 234}, {'id': 11, 'elements': [9, 10, 12], 'cost': 255}, {'id': 12, 'elements': [11, 13, 14, 15], 'cost': 220}, {'id': 13, 'elements': [5, 6], 'cost': 98}, {'id': 14, 'elements': [12, 13, 14], 'cost': 36}, {'id': 15, 'elements': [9, 11], 'cost': 38}, {'id': 16, 'elements': [3, 6], 'cost': 80}, {'id': 17, 'elements': [1, 4], 'cost': 18}, {'id': 18, 'elements': [5, 7, 8, 9], 'cost': 192}, {'id': 19, 'elements': [2, 3], 'cost': 30}, {'id': 20, 'elements': [12, 13, 15], 'cost': 294}, {'id': 21, 'elements': [14, 15], 'cost': 160}, {'id': 22, 'elements': [10, 11], 'cost': 98}, {'id': 23, 'elements': [7, 10, 12], 'cost': 3}, {'id': 24, 'elements': [11, 13, 15], 'cost': 3}, {'id': 25, 'elements': [10, 11, 12], 'cost': 183}, {'id': 26, 'elements': [9, 10, 12, 15], 'cost': 124}, {'id': 27, 'elements': [1, 2, 3, 5], 'cost': 72}, {'id': 28, 'elements': [10, 13, 14], 'cost': 126}, {'id': 29, 'elements': [8, 9], 'cost': 190}, {'id': 30, 'elements': [7, 8], 'cost': 190}, {'id': 31, 'elements': [2, 4, 5], 'cost': 183}, {'id': 32, 'elements': [11, 14], 'cost': 142}, {'id': 33, 'elements': [6, 7, 8, 10], 'cost': 124}, {'id': 34, 'elements': [3, 4], 'cost': 24}, {'id': 35, 'elements': [9, 14], 'cost': 84}, {'id': 36, 'elements': [8, 9, 11, 12], 'cost': 356}, {'id': 37, 'elements': [10, 12, 13], 'cost': 279}, {'id': 38, 'elements': [10, 12, 14], 'cost': 90}, {'id': 39, 'elements': [1, 2], 'cost': 88}, {'id': 40, 'elements': [3, 7], 'cost': 16}, {'id': 41, 'elements': [3, 6, 7], 'cost': 141}, {'id': 42, 'elements': [8, 10, 11], 'cost': 114}, {'id': 43, 'elements': [6, 7, 9, 10], 'cost': 360}, {'id': 44, 'elements': [5, 10], 'cost': 32}, {'id': 45, 'elements': [13, 14], 'cost': 56}, {'id': 46, 'elements': [10, 13, 15], 'cost': 60}, {'id': 47, 'elements': [3, 4, 5, 6], 'cost': 400}, {'id': 48, 'elements': [7, 9], 'cost': 170}, {'id': 49, 'elements': [2, 3, 4, 7], 'cost': 180}, {'id': 50, 'elements': [5, 6, 8, 10], 'cost': 360}, {'id': 51, 'elements': [1, 3, 5, 6], 'cost': 372}, {'id': 52, 'elements': [11, 12, 14], 'cost': 183}, {'id': 53, 'elements': [6, 7, 10], 'cost': 15}, {'id': 54, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 66, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 67, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 68, 'elements': [15], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0025_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0025_bag_stats.png'}","[2, 13, 17, 19, 24, 38]",248.0,"{'num_elements': 15, 'num_sets': 68, 'sets': [{'id': 'S1', 'elements': [4, 6, 7], 'cost': 195}, {'id': 'S2', 'elements': [7, 8, 9], 'cost': 9}, {'id': 'S3', 'elements': [6, 7, 8, 9], 'cost': 80}, {'id': 'S4', 'elements': [2, 3, 4], 'cost': 147}, {'id': 'S5', 'elements': [6, 7, 9], 'cost': 60}, {'id': 'S6', 'elements': [1, 2, 3, 4], 'cost': 216}, {'id': 'S7', 'elements': [11, 12, 13, 14], 'cost': 212}, {'id': 'S8', 'elements': [12, 13, 14, 15], 'cost': 132}, {'id': 'S9', 'elements': [9, 10, 11, 12], 'cost': 188}, {'id': 'S10', 'elements': [8, 9, 12], 'cost': 234}, {'id': 'S11', 'elements': [9, 10, 12], 'cost': 255}, {'id': 'S12', 'elements': [11, 13, 14, 15], 'cost': 220}, {'id': 'S13', 'elements': [5, 6], 'cost': 98}, {'id': 'S14', 'elements': [12, 13, 14], 'cost': 36}, {'id': 'S15', 'elements': [9, 11], 'cost': 38}, {'id': 'S16', 'elements': [3, 6], 'cost': 80}, {'id': 'S17', 'elements': [1, 4], 'cost': 18}, {'id': 'S18', 'elements': [5, 7, 8, 9], 'cost': 192}, {'id': 'S19', 'elements': [2, 3], 'cost': 30}, {'id': 'S20', 'elements': [12, 13, 15], 'cost': 294}, {'id': 'S21', 'elements': [14, 15], 'cost': 160}, {'id': 'S22', 'elements': [10, 11], 'cost': 98}, {'id': 'S23', 'elements': [7, 10, 12], 'cost': 3}, {'id': 'S24', 'elements': [11, 13, 15], 'cost': 3}, {'id': 'S25', 'elements': [10, 11, 12], 'cost': 183}, {'id': 'S26', 'elements': [9, 10, 12, 15], 'cost': 124}, {'id': 'S27', 'elements': [1, 2, 3, 5], 'cost': 72}, {'id': 'S28', 'elements': [10, 13, 14], 'cost': 126}, {'id': 'S29', 'elements': [8, 9], 'cost': 190}, {'id': 'S30', 'elements': [7, 8], 'cost': 190}, {'id': 'S31', 'elements': [2, 4, 5], 'cost': 183}, {'id': 'S32', 'elements': [11, 14], 'cost': 142}, {'id': 'S33', 'elements': [6, 7, 8, 10], 'cost': 124}, {'id': 'S34', 'elements': [3, 4], 'cost': 24}, {'id': 'S35', 'elements': [9, 14], 'cost': 84}, {'id': 'S36', 'elements': [8, 9, 11, 12], 'cost': 356}, {'id': 'S37', 'elements': [10, 12, 13], 'cost': 279}, {'id': 'S38', 'elements': [10, 12, 14], 'cost': 90}, {'id': 'S39', 'elements': [1, 2], 'cost': 88}, {'id': 'S40', 'elements': [3, 7], 'cost': 16}, {'id': 'S41', 'elements': [3, 6, 7], 'cost': 141}, {'id': 'S42', 'elements': [8, 10, 11], 'cost': 114}, {'id': 'S43', 'elements': [6, 7, 9, 10], 'cost': 360}, {'id': 'S44', 'elements': [5, 10], 'cost': 32}, {'id': 'S45', 'elements': [13, 14], 'cost': 56}, {'id': 'S46', 'elements': [10, 13, 15], 'cost': 60}, {'id': 'S47', 'elements': [3, 4, 5, 6], 'cost': 400}, {'id': 'S48', 'elements': [7, 9], 'cost': 170}, {'id': 'S49', 'elements': [2, 3, 4, 7], 'cost': 180}, {'id': 'S50', 'elements': [5, 6, 8, 10], 'cost': 360}, {'id': 'S51', 'elements': [1, 3, 5, 6], 'cost': 372}, {'id': 'S52', 'elements': [11, 12, 14], 'cost': 183}, {'id': 'S53', 'elements': [6, 7, 10], 'cost': 15}, {'id': 'S54', 'elements': [1], 'cost': 10000}, {'id': 'S55', 'elements': [2], 'cost': 10000}, {'id': 'S56', 'elements': [3], 'cost': 10000}, {'id': 'S57', 'elements': [4], 'cost': 10000}, {'id': 'S58', 'elements': [5], 'cost': 10000}, {'id': 'S59', 'elements': [6], 'cost': 10000}, {'id': 'S60', 'elements': [7], 'cost': 10000}, {'id': 'S61', 'elements': [8], 'cost': 10000}, {'id': 'S62', 'elements': [9], 'cost': 10000}, {'id': 'S63', 'elements': [10], 'cost': 10000}, {'id': 'S64', 'elements': [11], 'cost': 10000}, {'id': 'S65', 'elements': [12], 'cost': 10000}, {'id': 'S66', 'elements': [13], 'cost': 10000}, {'id': 'S67', 'elements': [14], 'cost': 10000}, {'id': 'S68', 'elements': [15], 'cost': 10000}]}","['S2', 'S13', 'S17', 'S19', 'S24', 'S38']",26,csv,1
SPP,SPP,"I’m juggling a menu and a pile of pre-packed ingredient kits: the job is to pick which kits to order so every dish on the menu is made from exactly one of those kits — nothing left without a kit, and nothing made from two kits at once. The better choices are the ones that let all dishes be covered while keeping the bill as low as possible: total cost here means adding up the prices of the kits we actually order, and the goal is to make that sum as small as possible. The concrete details are listed below.

# num_dishes=14
# num_kits_available=61
kit_id,kit_price,dishes_in_kit
S1,24,12 14
S2,8,11 12 13 14
S3,132,1 3
S4,72,10 11
S5,249,1 2 3
S6,207,3 4 5
S7,138,13 14
S8,336,8 9 10 11
S9,252,10 12 13
S10,22,8 10
S11,340,7 8 11 12
S12,67,3
S13,268,4 5 6 7
S14,24,5 6 7 8
S15,56,3 6
S16,220,9 10 11 12
S17,42,4 6
S18,164,2 5
S19,168,2 3 7
S20,138,9 10
S21,24,8 9
S22,58,4 7
S23,135,3 5 6
S24,21,11 12 13
S25,190,12 13
S26,186,12 13 14
S27,51,2 3 5
S28,186,8 11
S29,273,5 6 7
S30,108,8 9 11
S31,48,7 9 10 11
S32,124,4 7 8 9
S33,8,1 2 3 6
S34,352,5 7 8 9
S35,384,8 9 10 12
S36,96,8
S37,392,6 7 9 10
S38,138,2 3 4
S39,348,5 6 7 10
S40,190,7 8
S41,249,6 7 10
S42,92,1 2
S43,92,5 8
S44,11,7
S45,297,1 3 5
S46,182,2 7
S47,108,2 3
S48,10000,1
S49,10000,2
S50,10000,3
S51,10000,4
S52,10000,5
S53,10000,6
S54,10000,7
S55,10000,8
S56,10000,9
S57,10000,10
S58,10000,11
S59,10000,12
S60,10000,13
S61,10000,14

Also, when you send your pick, please use this little JSON shape so it's easy to read and check. It's just a sketch of what I expect back:

{
  ""solution"": [""kit_id"", ...]
}

Here ""solution"" is a list of the kits you want to order — each entry is the identifier for one pre-packed kit that will cover a dish. Think of it like ticking boxes on a form: put the exact kit IDs you choose into that array. This JSON is just the shape I want, not the actual final answer.

One more thing: use the identifiers exactly as they appear in the instance input — do not rename them or invent new labels. Valid identifiers look like plain numbers such as ""1"" or ""23"", single capital letters like ""A"" or ""B"", or a capital letter followed by digits like ""A1"" or ""X7"".","{'problem_type': 'SPP', 'num_elements': 14, 'num_sets': 61, 'density': 0.1686182669789227, 'sets': [{'id': 1, 'elements': [12, 14], 'cost': 24}, {'id': 2, 'elements': [11, 12, 13, 14], 'cost': 8}, {'id': 3, 'elements': [1, 3], 'cost': 132}, {'id': 4, 'elements': [10, 11], 'cost': 72}, {'id': 5, 'elements': [1, 2, 3], 'cost': 249}, {'id': 6, 'elements': [3, 4, 5], 'cost': 207}, {'id': 7, 'elements': [13, 14], 'cost': 138}, {'id': 8, 'elements': [8, 9, 10, 11], 'cost': 336}, {'id': 9, 'elements': [10, 12, 13], 'cost': 252}, {'id': 10, 'elements': [8, 10], 'cost': 22}, {'id': 11, 'elements': [7, 8, 11, 12], 'cost': 340}, {'id': 12, 'elements': [3], 'cost': 67}, {'id': 13, 'elements': [4, 5, 6, 7], 'cost': 268}, {'id': 14, 'elements': [5, 6, 7, 8], 'cost': 24}, {'id': 15, 'elements': [3, 6], 'cost': 56}, {'id': 16, 'elements': [9, 10, 11, 12], 'cost': 220}, {'id': 17, 'elements': [4, 6], 'cost': 42}, {'id': 18, 'elements': [2, 5], 'cost': 164}, {'id': 19, 'elements': [2, 3, 7], 'cost': 168}, {'id': 20, 'elements': [9, 10], 'cost': 138}, {'id': 21, 'elements': [8, 9], 'cost': 24}, {'id': 22, 'elements': [4, 7], 'cost': 58}, {'id': 23, 'elements': [3, 5, 6], 'cost': 135}, {'id': 24, 'elements': [11, 12, 13], 'cost': 21}, {'id': 25, 'elements': [12, 13], 'cost': 190}, {'id': 26, 'elements': [12, 13, 14], 'cost': 186}, {'id': 27, 'elements': [2, 3, 5], 'cost': 51}, {'id': 28, 'elements': [8, 11], 'cost': 186}, {'id': 29, 'elements': [5, 6, 7], 'cost': 273}, {'id': 30, 'elements': [8, 9, 11], 'cost': 108}, {'id': 31, 'elements': [7, 9, 10, 11], 'cost': 48}, {'id': 32, 'elements': [4, 7, 8, 9], 'cost': 124}, {'id': 33, 'elements': [1, 2, 3, 6], 'cost': 8}, {'id': 34, 'elements': [5, 7, 8, 9], 'cost': 352}, {'id': 35, 'elements': [8, 9, 10, 12], 'cost': 384}, {'id': 36, 'elements': [8], 'cost': 96}, {'id': 37, 'elements': [6, 7, 9, 10], 'cost': 392}, {'id': 38, 'elements': [2, 3, 4], 'cost': 138}, {'id': 39, 'elements': [5, 6, 7, 10], 'cost': 348}, {'id': 40, 'elements': [7, 8], 'cost': 190}, {'id': 41, 'elements': [6, 7, 10], 'cost': 249}, {'id': 42, 'elements': [1, 2], 'cost': 92}, {'id': 43, 'elements': [5, 8], 'cost': 92}, {'id': 44, 'elements': [7], 'cost': 11}, {'id': 45, 'elements': [1, 3, 5], 'cost': 297}, {'id': 46, 'elements': [2, 7], 'cost': 182}, {'id': 47, 'elements': [2, 3], 'cost': 108}, {'id': 48, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [14], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0026_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0026_bag_stats.png'}","[2, 20, 22, 33, 43]",304.0,"{'num_elements': 14, 'num_sets': 61, 'sets': [{'id': 'S1', 'elements': [12, 14], 'cost': 24}, {'id': 'S2', 'elements': [11, 12, 13, 14], 'cost': 8}, {'id': 'S3', 'elements': [1, 3], 'cost': 132}, {'id': 'S4', 'elements': [10, 11], 'cost': 72}, {'id': 'S5', 'elements': [1, 2, 3], 'cost': 249}, {'id': 'S6', 'elements': [3, 4, 5], 'cost': 207}, {'id': 'S7', 'elements': [13, 14], 'cost': 138}, {'id': 'S8', 'elements': [8, 9, 10, 11], 'cost': 336}, {'id': 'S9', 'elements': [10, 12, 13], 'cost': 252}, {'id': 'S10', 'elements': [8, 10], 'cost': 22}, {'id': 'S11', 'elements': [7, 8, 11, 12], 'cost': 340}, {'id': 'S12', 'elements': [3], 'cost': 67}, {'id': 'S13', 'elements': [4, 5, 6, 7], 'cost': 268}, {'id': 'S14', 'elements': [5, 6, 7, 8], 'cost': 24}, {'id': 'S15', 'elements': [3, 6], 'cost': 56}, {'id': 'S16', 'elements': [9, 10, 11, 12], 'cost': 220}, {'id': 'S17', 'elements': [4, 6], 'cost': 42}, {'id': 'S18', 'elements': [2, 5], 'cost': 164}, {'id': 'S19', 'elements': [2, 3, 7], 'cost': 168}, {'id': 'S20', 'elements': [9, 10], 'cost': 138}, {'id': 'S21', 'elements': [8, 9], 'cost': 24}, {'id': 'S22', 'elements': [4, 7], 'cost': 58}, {'id': 'S23', 'elements': [3, 5, 6], 'cost': 135}, {'id': 'S24', 'elements': [11, 12, 13], 'cost': 21}, {'id': 'S25', 'elements': [12, 13], 'cost': 190}, {'id': 'S26', 'elements': [12, 13, 14], 'cost': 186}, {'id': 'S27', 'elements': [2, 3, 5], 'cost': 51}, {'id': 'S28', 'elements': [8, 11], 'cost': 186}, {'id': 'S29', 'elements': [5, 6, 7], 'cost': 273}, {'id': 'S30', 'elements': [8, 9, 11], 'cost': 108}, {'id': 'S31', 'elements': [7, 9, 10, 11], 'cost': 48}, {'id': 'S32', 'elements': [4, 7, 8, 9], 'cost': 124}, {'id': 'S33', 'elements': [1, 2, 3, 6], 'cost': 8}, {'id': 'S34', 'elements': [5, 7, 8, 9], 'cost': 352}, {'id': 'S35', 'elements': [8, 9, 10, 12], 'cost': 384}, {'id': 'S36', 'elements': [8], 'cost': 96}, {'id': 'S37', 'elements': [6, 7, 9, 10], 'cost': 392}, {'id': 'S38', 'elements': [2, 3, 4], 'cost': 138}, {'id': 'S39', 'elements': [5, 6, 7, 10], 'cost': 348}, {'id': 'S40', 'elements': [7, 8], 'cost': 190}, {'id': 'S41', 'elements': [6, 7, 10], 'cost': 249}, {'id': 'S42', 'elements': [1, 2], 'cost': 92}, {'id': 'S43', 'elements': [5, 8], 'cost': 92}, {'id': 'S44', 'elements': [7], 'cost': 11}, {'id': 'S45', 'elements': [1, 3, 5], 'cost': 297}, {'id': 'S46', 'elements': [2, 7], 'cost': 182}, {'id': 'S47', 'elements': [2, 3], 'cost': 108}, {'id': 'S48', 'elements': [1], 'cost': 10000}, {'id': 'S49', 'elements': [2], 'cost': 10000}, {'id': 'S50', 'elements': [3], 'cost': 10000}, {'id': 'S51', 'elements': [4], 'cost': 10000}, {'id': 'S52', 'elements': [5], 'cost': 10000}, {'id': 'S53', 'elements': [6], 'cost': 10000}, {'id': 'S54', 'elements': [7], 'cost': 10000}, {'id': 'S55', 'elements': [8], 'cost': 10000}, {'id': 'S56', 'elements': [9], 'cost': 10000}, {'id': 'S57', 'elements': [10], 'cost': 10000}, {'id': 'S58', 'elements': [11], 'cost': 10000}, {'id': 'S59', 'elements': [12], 'cost': 10000}, {'id': 'S60', 'elements': [13], 'cost': 10000}, {'id': 'S61', 'elements': [14], 'cost': 10000}]}","['S2', 'S20', 'S22', 'S33', 'S43']",27,csv,1
SPP,SPP,"There’s an itinerary with a bunch of attractions and several overlapping tour options, so the trick is choosing tours so each attraction is handled by exactly one of the tours bought. After selecting a combination, the cost is just the sum of the prices of those tours, and the aim is to make that sum as small as possible while making sure nothing is uncovered or double-booked. The exact tour and attraction list appears below.

- **total_attractions_count**: 18
- **total_tour_packages**: 87

| tour_package_id | tour_price | attractions_covered |
|---|---|---|
| S1 | 150 | 13 15 16 |
| S2 | 90 | 2 5 |
| S3 | 8 | 5 8 9 10 |
| S4 | 240 | 7 9 10 |
| S5 | 30 | 15 16 |
| S6 | 470 | 5 7 8 9 10 |
| S7 | 64 | 4 5 6 8 |
| S8 | 38 | 1 2 |
| S9 | 120 | 4 5 7 |
| S10 | 28 | 5 7 8 9 |
| S11 | 96 | 8 10 12 |
| S12 | 126 | 1 2 4 |
| S13 | 99 | 15 16 17 |
| S14 | 85 | 10 11 12 13 14 |
| S15 | 93 | 11 12 13 |
| S16 | 312 | 4 5 7 8 |
| S17 | 26 | 5 7 |
| S18 | 122 | 16 17 |
| S19 | 490 | 5 6 7 8 9 |
| S20 | 45 | 14 15 16 17 18 |
| S21 | 360 | 13 16 17 18 |
| S22 | 255 | 7 9 10 11 12 |
| S23 | 200 | 15 16 17 18 |
| S24 | 112 | 10 11 12 13 |
| S25 | 272 | 1 2 3 4 |
| S26 | 87 | 5 7 8 |
| S27 | 72 | 7 9 |
| S28 | 75 | 6 8 11 12 13 |
| S29 | 84 | 13 16 |
| S30 | 180 | 7 8 9 11 12 |
| S31 | 390 | 2 3 4 5 7 |
| S32 | 388 | 14 15 16 17 |
| S33 | 186 | 14 15 16 |
| S34 | 46 | 6 8 |
| S35 | 74 | 14 18 |
| S36 | 176 | 7 8 |
| S37 | 208 | 2 3 4 5 |
| S38 | 4 | 15 18 |
| S39 | 24 | 9 11 12 15 |
| S40 | 154 | 2 6 |
| S41 | 190 | 1 4 |
| S42 | 215 | 1 2 3 4 5 |
| S43 | 24 | 5 6 8 9 |
| S44 | 108 | 8 12 |
| S45 | 38 | 4 8 |
| S46 | 276 | 9 10 12 |
| S47 | 50 | 2 3 4 7 8 |
| S48 | 80 | 1 2 3 7 |
| S49 | 105 | 9 11 13 |
| S50 | 100 | 7 9 12 13 15 |
| S51 | 45 | 3 4 5 6 7 |
| S52 | 24 | 13 17 |
| S53 | 165 | 3 4 5 |
| S54 | 128 | 12 14 15 17 |
| S55 | 90 | 16 17 18 |
| S56 | 9 | 1 2 8 |
| S57 | 135 | 13 17 18 |
| S58 | 162 | 3 4 7 |
| S59 | 273 | 5 6 11 |
| S60 | 14 | 1 3 |
| S61 | 108 | 5 8 |
| S62 | 136 | 4 6 |
| S63 | 8 | 9 10 11 12 |
| S64 | 69 | 2 5 6 |
| S65 | 368 | 1 2 3 5 |
| S66 | 245 | 13 15 16 17 18 |
| S67 | 360 | 2 4 5 6 8 |
| S68 | 185 | 11 13 15 16 18 |
| S69 | 130 | 14 16 |
| S70 | 10000 | 1 |
| S71 | 10000 | 2 |
| S72 | 10000 | 3 |
| S73 | 10000 | 4 |
| S74 | 10000 | 5 |
| S75 | 10000 | 6 |
| S76 | 10000 | 7 |
| S77 | 10000 | 8 |
| S78 | 10000 | 9 |
| S79 | 10000 | 10 |
| S80 | 10000 | 11 |
| S81 | 10000 | 12 |
| S82 | 10000 | 13 |
| S83 | 10000 | 14 |
| S84 | 10000 | 15 |
| S85 | 10000 | 16 |
| S86 | 10000 | 17 |
| S87 | 10000 | 18 |

If you want to send the picks back, just use this simple JSON shape — nothing fancy, just the expected layout.

{
  ""solution"": [""tour_id"", ...]
}

""solution"" is an array that should list the IDs of the tours you choose (one entry per chosen tour). The ""tour_id"" item is just a placeholder showing the kind of values to put in there. This is only a sketch of the expected shape — not the actual answer.

Please make sure to use the exact identifiers from the instance input, with no renaming or extra labels.  
Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'problem_type': 'SPP', 'num_elements': 18, 'num_sets': 87, 'density': 0.1602809706257982, 'sets': [{'id': 1, 'elements': [13, 15, 16], 'cost': 150}, {'id': 2, 'elements': [2, 5], 'cost': 90}, {'id': 3, 'elements': [5, 8, 9, 10], 'cost': 8}, {'id': 4, 'elements': [7, 9, 10], 'cost': 240}, {'id': 5, 'elements': [15, 16], 'cost': 30}, {'id': 6, 'elements': [5, 7, 8, 9, 10], 'cost': 470}, {'id': 7, 'elements': [4, 5, 6, 8], 'cost': 64}, {'id': 8, 'elements': [1, 2], 'cost': 38}, {'id': 9, 'elements': [4, 5, 7], 'cost': 120}, {'id': 10, 'elements': [5, 7, 8, 9], 'cost': 28}, {'id': 11, 'elements': [8, 10, 12], 'cost': 96}, {'id': 12, 'elements': [1, 2, 4], 'cost': 126}, {'id': 13, 'elements': [15, 16, 17], 'cost': 99}, {'id': 14, 'elements': [10, 11, 12, 13, 14], 'cost': 85}, {'id': 15, 'elements': [11, 12, 13], 'cost': 93}, {'id': 16, 'elements': [4, 5, 7, 8], 'cost': 312}, {'id': 17, 'elements': [5, 7], 'cost': 26}, {'id': 18, 'elements': [16, 17], 'cost': 122}, {'id': 19, 'elements': [5, 6, 7, 8, 9], 'cost': 490}, {'id': 20, 'elements': [14, 15, 16, 17, 18], 'cost': 45}, {'id': 21, 'elements': [13, 16, 17, 18], 'cost': 360}, {'id': 22, 'elements': [7, 9, 10, 11, 12], 'cost': 255}, {'id': 23, 'elements': [15, 16, 17, 18], 'cost': 200}, {'id': 24, 'elements': [10, 11, 12, 13], 'cost': 112}, {'id': 25, 'elements': [1, 2, 3, 4], 'cost': 272}, {'id': 26, 'elements': [5, 7, 8], 'cost': 87}, {'id': 27, 'elements': [7, 9], 'cost': 72}, {'id': 28, 'elements': [6, 8, 11, 12, 13], 'cost': 75}, {'id': 29, 'elements': [13, 16], 'cost': 84}, {'id': 30, 'elements': [7, 8, 9, 11, 12], 'cost': 180}, {'id': 31, 'elements': [2, 3, 4, 5, 7], 'cost': 390}, {'id': 32, 'elements': [14, 15, 16, 17], 'cost': 388}, {'id': 33, 'elements': [14, 15, 16], 'cost': 186}, {'id': 34, 'elements': [6, 8], 'cost': 46}, {'id': 35, 'elements': [14, 18], 'cost': 74}, {'id': 36, 'elements': [7, 8], 'cost': 176}, {'id': 37, 'elements': [2, 3, 4, 5], 'cost': 208}, {'id': 38, 'elements': [15, 18], 'cost': 4}, {'id': 39, 'elements': [9, 11, 12, 15], 'cost': 24}, {'id': 40, 'elements': [2, 6], 'cost': 154}, {'id': 41, 'elements': [1, 4], 'cost': 190}, {'id': 42, 'elements': [1, 2, 3, 4, 5], 'cost': 215}, {'id': 43, 'elements': [5, 6, 8, 9], 'cost': 24}, {'id': 44, 'elements': [8, 12], 'cost': 108}, {'id': 45, 'elements': [4, 8], 'cost': 38}, {'id': 46, 'elements': [9, 10, 12], 'cost': 276}, {'id': 47, 'elements': [2, 3, 4, 7, 8], 'cost': 50}, {'id': 48, 'elements': [1, 2, 3, 7], 'cost': 80}, {'id': 49, 'elements': [9, 11, 13], 'cost': 105}, {'id': 50, 'elements': [7, 9, 12, 13, 15], 'cost': 100}, {'id': 51, 'elements': [3, 4, 5, 6, 7], 'cost': 45}, {'id': 52, 'elements': [13, 17], 'cost': 24}, {'id': 53, 'elements': [3, 4, 5], 'cost': 165}, {'id': 54, 'elements': [12, 14, 15, 17], 'cost': 128}, {'id': 55, 'elements': [16, 17, 18], 'cost': 90}, {'id': 56, 'elements': [1, 2, 8], 'cost': 9}, {'id': 57, 'elements': [13, 17, 18], 'cost': 135}, {'id': 58, 'elements': [3, 4, 7], 'cost': 162}, {'id': 59, 'elements': [5, 6, 11], 'cost': 273}, {'id': 60, 'elements': [1, 3], 'cost': 14}, {'id': 61, 'elements': [5, 8], 'cost': 108}, {'id': 62, 'elements': [4, 6], 'cost': 136}, {'id': 63, 'elements': [9, 10, 11, 12], 'cost': 8}, {'id': 64, 'elements': [2, 5, 6], 'cost': 69}, {'id': 65, 'elements': [1, 2, 3, 5], 'cost': 368}, {'id': 66, 'elements': [13, 15, 16, 17, 18], 'cost': 245}, {'id': 67, 'elements': [2, 4, 5, 6, 8], 'cost': 360}, {'id': 68, 'elements': [11, 13, 15, 16, 18], 'cost': 185}, {'id': 69, 'elements': [14, 16], 'cost': 130}, {'id': 70, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 71, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 72, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 73, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 74, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 75, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 76, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 77, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 78, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 79, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 80, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 81, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 82, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 83, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 84, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 85, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 86, 'elements': [17], 'cost': 10000, 'singleton': True}, {'id': 87, 'elements': [18], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0027_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0027_bag_stats.png'}","[5, 35, 51, 52, 56, 63]",190.0,"{'num_elements': 18, 'num_sets': 87, 'sets': [{'id': 'S1', 'elements': [13, 15, 16], 'cost': 150}, {'id': 'S2', 'elements': [2, 5], 'cost': 90}, {'id': 'S3', 'elements': [5, 8, 9, 10], 'cost': 8}, {'id': 'S4', 'elements': [7, 9, 10], 'cost': 240}, {'id': 'S5', 'elements': [15, 16], 'cost': 30}, {'id': 'S6', 'elements': [5, 7, 8, 9, 10], 'cost': 470}, {'id': 'S7', 'elements': [4, 5, 6, 8], 'cost': 64}, {'id': 'S8', 'elements': [1, 2], 'cost': 38}, {'id': 'S9', 'elements': [4, 5, 7], 'cost': 120}, {'id': 'S10', 'elements': [5, 7, 8, 9], 'cost': 28}, {'id': 'S11', 'elements': [8, 10, 12], 'cost': 96}, {'id': 'S12', 'elements': [1, 2, 4], 'cost': 126}, {'id': 'S13', 'elements': [15, 16, 17], 'cost': 99}, {'id': 'S14', 'elements': [10, 11, 12, 13, 14], 'cost': 85}, {'id': 'S15', 'elements': [11, 12, 13], 'cost': 93}, {'id': 'S16', 'elements': [4, 5, 7, 8], 'cost': 312}, {'id': 'S17', 'elements': [5, 7], 'cost': 26}, {'id': 'S18', 'elements': [16, 17], 'cost': 122}, {'id': 'S19', 'elements': [5, 6, 7, 8, 9], 'cost': 490}, {'id': 'S20', 'elements': [14, 15, 16, 17, 18], 'cost': 45}, {'id': 'S21', 'elements': [13, 16, 17, 18], 'cost': 360}, {'id': 'S22', 'elements': [7, 9, 10, 11, 12], 'cost': 255}, {'id': 'S23', 'elements': [15, 16, 17, 18], 'cost': 200}, {'id': 'S24', 'elements': [10, 11, 12, 13], 'cost': 112}, {'id': 'S25', 'elements': [1, 2, 3, 4], 'cost': 272}, {'id': 'S26', 'elements': [5, 7, 8], 'cost': 87}, {'id': 'S27', 'elements': [7, 9], 'cost': 72}, {'id': 'S28', 'elements': [6, 8, 11, 12, 13], 'cost': 75}, {'id': 'S29', 'elements': [13, 16], 'cost': 84}, {'id': 'S30', 'elements': [7, 8, 9, 11, 12], 'cost': 180}, {'id': 'S31', 'elements': [2, 3, 4, 5, 7], 'cost': 390}, {'id': 'S32', 'elements': [14, 15, 16, 17], 'cost': 388}, {'id': 'S33', 'elements': [14, 15, 16], 'cost': 186}, {'id': 'S34', 'elements': [6, 8], 'cost': 46}, {'id': 'S35', 'elements': [14, 18], 'cost': 74}, {'id': 'S36', 'elements': [7, 8], 'cost': 176}, {'id': 'S37', 'elements': [2, 3, 4, 5], 'cost': 208}, {'id': 'S38', 'elements': [15, 18], 'cost': 4}, {'id': 'S39', 'elements': [9, 11, 12, 15], 'cost': 24}, {'id': 'S40', 'elements': [2, 6], 'cost': 154}, {'id': 'S41', 'elements': [1, 4], 'cost': 190}, {'id': 'S42', 'elements': [1, 2, 3, 4, 5], 'cost': 215}, {'id': 'S43', 'elements': [5, 6, 8, 9], 'cost': 24}, {'id': 'S44', 'elements': [8, 12], 'cost': 108}, {'id': 'S45', 'elements': [4, 8], 'cost': 38}, {'id': 'S46', 'elements': [9, 10, 12], 'cost': 276}, {'id': 'S47', 'elements': [2, 3, 4, 7, 8], 'cost': 50}, {'id': 'S48', 'elements': [1, 2, 3, 7], 'cost': 80}, {'id': 'S49', 'elements': [9, 11, 13], 'cost': 105}, {'id': 'S50', 'elements': [7, 9, 12, 13, 15], 'cost': 100}, {'id': 'S51', 'elements': [3, 4, 5, 6, 7], 'cost': 45}, {'id': 'S52', 'elements': [13, 17], 'cost': 24}, {'id': 'S53', 'elements': [3, 4, 5], 'cost': 165}, {'id': 'S54', 'elements': [12, 14, 15, 17], 'cost': 128}, {'id': 'S55', 'elements': [16, 17, 18], 'cost': 90}, {'id': 'S56', 'elements': [1, 2, 8], 'cost': 9}, {'id': 'S57', 'elements': [13, 17, 18], 'cost': 135}, {'id': 'S58', 'elements': [3, 4, 7], 'cost': 162}, {'id': 'S59', 'elements': [5, 6, 11], 'cost': 273}, {'id': 'S60', 'elements': [1, 3], 'cost': 14}, {'id': 'S61', 'elements': [5, 8], 'cost': 108}, {'id': 'S62', 'elements': [4, 6], 'cost': 136}, {'id': 'S63', 'elements': [9, 10, 11, 12], 'cost': 8}, {'id': 'S64', 'elements': [2, 5, 6], 'cost': 69}, {'id': 'S65', 'elements': [1, 2, 3, 5], 'cost': 368}, {'id': 'S66', 'elements': [13, 15, 16, 17, 18], 'cost': 245}, {'id': 'S67', 'elements': [2, 4, 5, 6, 8], 'cost': 360}, {'id': 'S68', 'elements': [11, 13, 15, 16, 18], 'cost': 185}, {'id': 'S69', 'elements': [14, 16], 'cost': 130}, {'id': 'S70', 'elements': [1], 'cost': 10000}, {'id': 'S71', 'elements': [2], 'cost': 10000}, {'id': 'S72', 'elements': [3], 'cost': 10000}, {'id': 'S73', 'elements': [4], 'cost': 10000}, {'id': 'S74', 'elements': [5], 'cost': 10000}, {'id': 'S75', 'elements': [6], 'cost': 10000}, {'id': 'S76', 'elements': [7], 'cost': 10000}, {'id': 'S77', 'elements': [8], 'cost': 10000}, {'id': 'S78', 'elements': [9], 'cost': 10000}, {'id': 'S79', 'elements': [10], 'cost': 10000}, {'id': 'S80', 'elements': [11], 'cost': 10000}, {'id': 'S81', 'elements': [12], 'cost': 10000}, {'id': 'S82', 'elements': [13], 'cost': 10000}, {'id': 'S83', 'elements': [14], 'cost': 10000}, {'id': 'S84', 'elements': [15], 'cost': 10000}, {'id': 'S85', 'elements': [16], 'cost': 10000}, {'id': 'S86', 'elements': [17], 'cost': 10000}, {'id': 'S87', 'elements': [18], 'cost': 10000}]}","['S5', 'S35', 'S51', 'S52', 'S56', 'S63']",28,markdown_table,1
SPP,SPP,"Many people on the floor have noticed the assortment of prepacked box types and are trying to figure out which ones to use for an upcoming shipment. Each box type holds particular product units and comes with a cost, so the decision is which box types to select so that every product unit is put into one—and only one—selected box type. The better plan is the cheaper plan: add up the costs of the chosen box types to get the total packing cost, and prefer the smallest total. The concrete box and unit details are shown below.

There are 14 product units and 51 prepacked box types listed below:
Box type S1 holds product units F G H J and costs 240.
Box type S2 holds product units I L and costs 56.
Box type S3 holds product units J K L N and costs 376.
Box type S4 holds product units H J K and costs 132.
Box type S5 holds product units I K and costs 184.
Box type S6 holds product units F H J K and costs 292.
Box type S7 holds product units E F I and costs 120.
Box type S8 holds product units K L N and costs 123.
Box type S9 holds product units D E and costs 94.
Box type S10 holds product units A C and costs 68.
Box type S11 holds product units D F G H and costs 292.
Box type S12 holds product units E G H and costs 264.
Box type S13 holds product units F G and costs 82.
Box type S14 holds product units J K and costs 176.
Box type S15 holds product units F H J and costs 63.
Box type S16 holds product units L M N and costs 36.
Box type S17 holds product units I J and costs 120.
Box type S18 holds product units J L M N and costs 312.
Box type S19 holds product units J L M and costs 210.
Box type S20 holds product units G H and costs 146.
Box type S21 holds product units K and costs 8.
Box type S22 holds product units D E G I and costs 12.
Box type S23 holds product units I J K L and costs 144.
Box type S24 holds product units D E G and costs 84.
Box type S25 holds product units C D G and costs 165.
Box type S26 holds product units K L and costs 108.
Box type S27 holds product units G I J and costs 138.
Box type S28 holds product units G I J K and costs 40.
Box type S29 holds product units I J M and costs 282.
Box type S30 holds product units I M N and costs 57.
Box type S31 holds product units D F I and costs 201.
Box type S32 holds product units H I J and costs 246.
Box type S33 holds product units A B C D and costs 400.
Box type S34 holds product units H K and costs 94.
Box type S35 holds product units E and costs 16.
Box type S36 holds product units J K N and costs 210.
Box type S37 holds product units M N and costs 164.
Box type S38 holds product units A and costs 10000.
Box type S39 holds product units B and costs 10000.
Box type S40 holds product units C and costs 10000.
Box type S41 holds product units D and costs 10000.
Box type S42 holds product units E and costs 10000.
Box type S43 holds product units F and costs 10000.
Box type S44 holds product units G and costs 10000.
Box type S45 holds product units H and costs 10000.
Box type S46 holds product units I and costs 10000.
Box type S47 holds product units J and costs 10000.
Box type S48 holds product units K and costs 10000.
Box type S49 holds product units L and costs 10000.
Box type S50 holds product units M and costs 10000.
Box type S51 holds product units N and costs 10000.
They will prefer a selection that covers all 14 units exactly once while minimizing the total cost.

If you want to tell me which box types to pick, just send it in this simple JSON shape:

{
  ""solution"": [""box_type_id"", ...]
}

This is just a tiny sketch of the form I expect: ""solution"" holds a list of the chosen box type identifiers (one identifier per chosen box type). Think of each entry as the label of a prepacked box type you'd like to use.

The JSON above is only the expected shape, not the actual answer — fill the array with the exact identifiers from the instance when you reply. Remember: all identifiers must be used exactly as they appear in the input — no renaming and no new labels.

- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 14, 'num_sets': 51, 'density': 0.16666666666666666, 'sets': [{'id': 1, 'elements': [6, 7, 8, 10], 'cost': 240}, {'id': 2, 'elements': [9, 12], 'cost': 56}, {'id': 3, 'elements': [10, 11, 12, 14], 'cost': 376}, {'id': 4, 'elements': [8, 10, 11], 'cost': 132}, {'id': 5, 'elements': [9, 11], 'cost': 184}, {'id': 6, 'elements': [6, 8, 10, 11], 'cost': 292}, {'id': 7, 'elements': [5, 6, 9], 'cost': 120}, {'id': 8, 'elements': [11, 12, 14], 'cost': 123}, {'id': 9, 'elements': [4, 5], 'cost': 94}, {'id': 10, 'elements': [1, 3], 'cost': 68}, {'id': 11, 'elements': [4, 6, 7, 8], 'cost': 292}, {'id': 12, 'elements': [5, 7, 8], 'cost': 264}, {'id': 13, 'elements': [6, 7], 'cost': 82}, {'id': 14, 'elements': [10, 11], 'cost': 176}, {'id': 15, 'elements': [6, 8, 10], 'cost': 63}, {'id': 16, 'elements': [12, 13, 14], 'cost': 36}, {'id': 17, 'elements': [9, 10], 'cost': 120}, {'id': 18, 'elements': [10, 12, 13, 14], 'cost': 312}, {'id': 19, 'elements': [10, 12, 13], 'cost': 210}, {'id': 20, 'elements': [7, 8], 'cost': 146}, {'id': 21, 'elements': [11], 'cost': 8}, {'id': 22, 'elements': [4, 5, 7, 9], 'cost': 12}, {'id': 23, 'elements': [9, 10, 11, 12], 'cost': 144}, {'id': 24, 'elements': [4, 5, 7], 'cost': 84}, {'id': 25, 'elements': [3, 4, 7], 'cost': 165}, {'id': 26, 'elements': [11, 12], 'cost': 108}, {'id': 27, 'elements': [7, 9, 10], 'cost': 138}, {'id': 28, 'elements': [7, 9, 10, 11], 'cost': 40}, {'id': 29, 'elements': [9, 10, 13], 'cost': 282}, {'id': 30, 'elements': [9, 13, 14], 'cost': 57}, {'id': 31, 'elements': [4, 6, 9], 'cost': 201}, {'id': 32, 'elements': [8, 9, 10], 'cost': 246}, {'id': 33, 'elements': [1, 2, 3, 4], 'cost': 400}, {'id': 34, 'elements': [8, 11], 'cost': 94}, {'id': 35, 'elements': [5], 'cost': 16}, {'id': 36, 'elements': [10, 11, 14], 'cost': 210}, {'id': 37, 'elements': [13, 14], 'cost': 164}, {'id': 38, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 39, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 40, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 41, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 42, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 43, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 44, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 45, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 46, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 47, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [14], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0028_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0028_bag_stats.png'}","[13, 16, 17, 33, 34, 35]",748.0,"{'num_elements': 14, 'num_sets': 51, 'sets': [{'id': 'S1', 'elements': ['F', 'G', 'H', 'J'], 'cost': 240}, {'id': 'S2', 'elements': ['I', 'L'], 'cost': 56}, {'id': 'S3', 'elements': ['J', 'K', 'L', 'N'], 'cost': 376}, {'id': 'S4', 'elements': ['H', 'J', 'K'], 'cost': 132}, {'id': 'S5', 'elements': ['I', 'K'], 'cost': 184}, {'id': 'S6', 'elements': ['F', 'H', 'J', 'K'], 'cost': 292}, {'id': 'S7', 'elements': ['E', 'F', 'I'], 'cost': 120}, {'id': 'S8', 'elements': ['K', 'L', 'N'], 'cost': 123}, {'id': 'S9', 'elements': ['D', 'E'], 'cost': 94}, {'id': 'S10', 'elements': ['A', 'C'], 'cost': 68}, {'id': 'S11', 'elements': ['D', 'F', 'G', 'H'], 'cost': 292}, {'id': 'S12', 'elements': ['E', 'G', 'H'], 'cost': 264}, {'id': 'S13', 'elements': ['F', 'G'], 'cost': 82}, {'id': 'S14', 'elements': ['J', 'K'], 'cost': 176}, {'id': 'S15', 'elements': ['F', 'H', 'J'], 'cost': 63}, {'id': 'S16', 'elements': ['L', 'M', 'N'], 'cost': 36}, {'id': 'S17', 'elements': ['I', 'J'], 'cost': 120}, {'id': 'S18', 'elements': ['J', 'L', 'M', 'N'], 'cost': 312}, {'id': 'S19', 'elements': ['J', 'L', 'M'], 'cost': 210}, {'id': 'S20', 'elements': ['G', 'H'], 'cost': 146}, {'id': 'S21', 'elements': ['K'], 'cost': 8}, {'id': 'S22', 'elements': ['D', 'E', 'G', 'I'], 'cost': 12}, {'id': 'S23', 'elements': ['I', 'J', 'K', 'L'], 'cost': 144}, {'id': 'S24', 'elements': ['D', 'E', 'G'], 'cost': 84}, {'id': 'S25', 'elements': ['C', 'D', 'G'], 'cost': 165}, {'id': 'S26', 'elements': ['K', 'L'], 'cost': 108}, {'id': 'S27', 'elements': ['G', 'I', 'J'], 'cost': 138}, {'id': 'S28', 'elements': ['G', 'I', 'J', 'K'], 'cost': 40}, {'id': 'S29', 'elements': ['I', 'J', 'M'], 'cost': 282}, {'id': 'S30', 'elements': ['I', 'M', 'N'], 'cost': 57}, {'id': 'S31', 'elements': ['D', 'F', 'I'], 'cost': 201}, {'id': 'S32', 'elements': ['H', 'I', 'J'], 'cost': 246}, {'id': 'S33', 'elements': ['A', 'B', 'C', 'D'], 'cost': 400}, {'id': 'S34', 'elements': ['H', 'K'], 'cost': 94}, {'id': 'S35', 'elements': ['E'], 'cost': 16}, {'id': 'S36', 'elements': ['J', 'K', 'N'], 'cost': 210}, {'id': 'S37', 'elements': ['M', 'N'], 'cost': 164}, {'id': 'S38', 'elements': ['A'], 'cost': 10000}, {'id': 'S39', 'elements': ['B'], 'cost': 10000}, {'id': 'S40', 'elements': ['C'], 'cost': 10000}, {'id': 'S41', 'elements': ['D'], 'cost': 10000}, {'id': 'S42', 'elements': ['E'], 'cost': 10000}, {'id': 'S43', 'elements': ['F'], 'cost': 10000}, {'id': 'S44', 'elements': ['G'], 'cost': 10000}, {'id': 'S45', 'elements': ['H'], 'cost': 10000}, {'id': 'S46', 'elements': ['I'], 'cost': 10000}, {'id': 'S47', 'elements': ['J'], 'cost': 10000}, {'id': 'S48', 'elements': ['K'], 'cost': 10000}, {'id': 'S49', 'elements': ['L'], 'cost': 10000}, {'id': 'S50', 'elements': ['M'], 'cost': 10000}, {'id': 'S51', 'elements': ['N'], 'cost': 10000}]}","['S13', 'S16', 'S17', 'S33', 'S34', 'S35']",29,nl,names
SPP,SPP,"Imagine needing to buy a set of compilation albums to cover a playlist where each song must be available from just one licensed compilation — that means no missing songs and no duplicates among the purchased albums. The winning plan is simply the one with the lowest total license cost, which is the sum of the fees for the selected albums. The full details on albums, which songs they include, and their prices are provided below.

{
  ""num_songs"": 13,
  ""num_albums"": 56,
  ""sets"": [
    {
      ""album_id"": ""S1"",
      ""album_songs"": [
        ""A"",
        ""B"",
        ""E""
      ],
      ""license_fee"": 42
    },
    {
      ""album_id"": ""S2"",
      ""album_songs"": [
        ""K"",
        ""L"",
        ""M""
      ],
      ""license_fee"": 195
    },
    {
      ""album_id"": ""S3"",
      ""album_songs"": [
        ""A"",
        ""B"",
        ""C""
      ],
      ""license_fee"": 300
    },
    {
      ""album_id"": ""S4"",
      ""album_songs"": [
        ""E"",
        ""H""
      ],
      ""license_fee"": 32
    },
    {
      ""album_id"": ""S5"",
      ""album_songs"": [
        ""L"",
        ""M""
      ],
      ""license_fee"": 166
    },
    {
      ""album_id"": ""S6"",
      ""album_songs"": [
        ""D"",
        ""F"",
        ""G""
      ],
      ""license_fee"": 162
    },
    {
      ""album_id"": ""S7"",
      ""album_songs"": [
        ""F"",
        ""G""
      ],
      ""license_fee"": 190
    },
    {
      ""album_id"": ""S8"",
      ""album_songs"": [
        ""E"",
        ""F"",
        ""G""
      ],
      ""license_fee"": 216
    },
    {
      ""album_id"": ""S9"",
      ""album_songs"": [
        ""B"",
        ""C"",
        ""D"",
        ""F""
      ],
      ""license_fee"": 220
    },
    {
      ""album_id"": ""S10"",
      ""album_songs"": [
        ""K"",
        ""L""
      ],
      ""license_fee"": 104
    },
    {
      ""album_id"": ""S11"",
      ""album_songs"": [
        ""G"",
        ""H"",
        ""I"",
        ""J""
      ],
      ""license_fee"": 64
    },
    {
      ""album_id"": ""S12"",
      ""album_songs"": [
        ""I"",
        ""L""
      ],
      ""license_fee"": 160
    },
    {
      ""album_id"": ""S13"",
      ""album_songs"": [
        ""A"",
        ""B"",
        ""D""
      ],
      ""license_fee"": 177
    },
    {
      ""album_id"": ""S14"",
      ""album_songs"": [
        ""I"",
        ""J""
      ],
      ""license_fee"": 98
    },
    {
      ""album_id"": ""S15"",
      ""album_songs"": [
        ""I"",
        ""J"",
        ""K""
      ],
      ""license_fee"": 9
    },
    {
      ""album_id"": ""S16"",
      ""album_songs"": [
        ""H"",
        ""K"",
        ""L""
      ],
      ""license_fee"": 48
    },
    {
      ""album_id"": ""S17"",
      ""album_songs"": [
        ""D"",
        ""E""
      ],
      ""license_fee"": 20
    },
    {
      ""album_id"": ""S18"",
      ""album_songs"": [
        ""D"",
        ""E"",
        ""H""
      ],
      ""license_fee"": 9
    },
    {
      ""album_id"": ""S19"",
      ""album_songs"": [
        ""F"",
        ""J""
      ],
      ""license_fee"": 176
    },
    {
      ""album_id"": ""S20"",
      ""album_songs"": [
        ""E"",
        ""F""
      ],
      ""license_fee"": 42
    },
    {
      ""album_id"": ""S21"",
      ""album_songs"": [
        ""B"",
        ""C""
      ],
      ""license_fee"": 162
    },
    {
      ""album_id"": ""S22"",
      ""album_songs"": [
        ""H"",
        ""J"",
        ""K"",
        ""L""
      ],
      ""license_fee"": 100
    },
    {
      ""album_id"": ""S23"",
      ""album_songs"": [
        ""F"",
        ""H"",
        ""I""
      ],
      ""license_fee"": 141
    },
    {
      ""album_id"": ""S24"",
      ""album_songs"": [
        ""K"",
        ""M""
      ],
      ""license_fee"": 124
    },
    {
      ""album_id"": ""S25"",
      ""album_songs"": [
        ""C"",
        ""D"",
        ""E"",
        ""F""
      ],
      ""license_fee"": 204
    },
    {
      ""album_id"": ""S26"",
      ""album_songs"": [
        ""E"",
        ""G"",
        ""H""
      ],
      ""license_fee"": 15
    },
    {
      ""album_id"": ""S27"",
      ""album_songs"": [
        ""D"",
        ""E"",
        ""F"",
        ""G""
      ],
      ""license_fee"": 272
    },
    {
      ""album_id"": ""S28"",
      ""album_songs"": [
        ""A"",
        ""C""
      ],
      ""license_fee"": 110
    },
    {
      ""album_id"": ""S29"",
      ""album_songs"": [
        ""E""
      ],
      ""license_fee"": 73
    },
    {
      ""album_id"": ""S30"",
      ""album_songs"": [
        ""C"",
        ""D"",
        ""E""
      ],
      ""license_fee"": 204
    },
    {
      ""album_id"": ""S31"",
      ""album_songs"": [
        ""H"",
        ""J""
      ],
      ""license_fee"": 172
    },
    {
      ""album_id"": ""S32"",
      ""album_songs"": [
        ""K""
      ],
      ""license_fee"": 97
    },
    {
      ""album_id"": ""S33"",
      ""album_songs"": [
        ""B""
      ],
      ""license_fee"": 74
    },
    {
      ""album_id"": ""S34"",
      ""album_songs"": [
        ""A"",
        ""B"",
        ""C"",
        ""D""
      ],
      ""license_fee"": 216
    },
    {
      ""album_id"": ""S35"",
      ""album_songs"": [
        ""G"",
        ""H"",
        ""I""
      ],
      ""license_fee"": 177
    },
    {
      ""album_id"": ""S36"",
      ""album_songs"": [
        ""I"",
        ""K"",
        ""M""
      ],
      ""license_fee"": 240
    },
    {
      ""album_id"": ""S37"",
      ""album_songs"": [
        ""A"",
        ""B""
      ],
      ""license_fee"": 28
    },
    {
      ""album_id"": ""S38"",
      ""album_songs"": [
        ""J"",
        ""K"",
        ""L"",
        ""M""
      ],
      ""license_fee"": 192
    },
    {
      ""album_id"": ""S39"",
      ""album_songs"": [
        ""J"",
        ""K"",
        ""L""
      ],
      ""license_fee"": 102
    },
    {
      ""album_id"": ""S40"",
      ""album_songs"": [
        ""A"",
        ""C"",
        ""D""
      ],
      ""license_fee"": 159
    },
    {
      ""album_id"": ""S41"",
      ""album_songs"": [
        ""D"",
        ""E"",
        ""F""
      ],
      ""license_fee"": 102
    },
    {
      ""album_id"": ""S42"",
      ""album_songs"": [
        ""J"",
        ""L"",
        ""M""
      ],
      ""license_fee"": 255
    },
    {
      ""album_id"": ""S43"",
      ""album_songs"": [
        ""B"",
        ""D"",
        ""E""
      ],
      ""license_fee"": 84
    },
    {
      ""album_id"": ""S44"",
      ""album_songs"": [
        ""A""
      ],
      ""license_fee"": 10000
    },
    {
      ""album_id"": ""S45"",
      ""album_songs"": [
        ""B""
      ],
      ""license_fee"": 10000
    },
    {
      ""album_id"": ""S46"",
      ""album_songs"": [
        ""C""
      ],
      ""license_fee"": 10000
    },
    {
      ""album_id"": ""S47"",
      ""album_songs"": [
        ""D""
      ],
      ""license_fee"": 10000
    },
    {
      ""album_id"": ""S48"",
      ""album_songs"": [
        ""E""
      ],
      ""license_fee"": 10000
    },
    {
      ""album_id"": ""S49"",
      ""album_songs"": [
        ""F""
      ],
      ""license_fee"": 10000
    },
    {
      ""album_id"": ""S50"",
      ""album_songs"": [
        ""G""
      ],
      ""license_fee"": 10000
    },
    {
      ""album_id"": ""S51"",
      ""album_songs"": [
        ""H""
      ],
      ""license_fee"": 10000
    },
    {
      ""album_id"": ""S52"",
      ""album_songs"": [
        ""I""
      ],
      ""license_fee"": 10000
    },
    {
      ""album_id"": ""S53"",
      ""album_songs"": [
        ""J""
      ],
      ""license_fee"": 10000
    },
    {
      ""album_id"": ""S54"",
      ""album_songs"": [
        ""K""
      ],
      ""license_fee"": 10000
    },
    {
      ""album_id"": ""S55"",
      ""album_songs"": [
        ""L""
      ],
      ""license_fee"": 10000
    },
    {
      ""album_id"": ""S56"",
      ""album_songs"": [
        ""M""
      ],
      ""license_fee"": 10000
    }
  ]
}

Oh, and when you send back the chosen albums, just stick to a tiny JSON snippet like this so it’s easy to parse:

{
  ""solution"": [""album_id""]
}

Think of the ""solution"" array as the shopping list — drop in the album identifiers you want to buy so every song is covered exactly once. This JSON is just the shape I expect, not the final answer itself.

Please use the exact identifiers from the instance input — no renaming and no invented labels. 
""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 13, 'num_sets': 56, 'density': 0.1771978021978022, 'sets': [{'id': 1, 'elements': [1, 2, 5], 'cost': 42}, {'id': 2, 'elements': [11, 12, 13], 'cost': 195}, {'id': 3, 'elements': [1, 2, 3], 'cost': 300}, {'id': 4, 'elements': [5, 8], 'cost': 32}, {'id': 5, 'elements': [12, 13], 'cost': 166}, {'id': 6, 'elements': [4, 6, 7], 'cost': 162}, {'id': 7, 'elements': [6, 7], 'cost': 190}, {'id': 8, 'elements': [5, 6, 7], 'cost': 216}, {'id': 9, 'elements': [2, 3, 4, 6], 'cost': 220}, {'id': 10, 'elements': [11, 12], 'cost': 104}, {'id': 11, 'elements': [7, 8, 9, 10], 'cost': 64}, {'id': 12, 'elements': [9, 12], 'cost': 160}, {'id': 13, 'elements': [1, 2, 4], 'cost': 177}, {'id': 14, 'elements': [9, 10], 'cost': 98}, {'id': 15, 'elements': [9, 10, 11], 'cost': 9}, {'id': 16, 'elements': [8, 11, 12], 'cost': 48}, {'id': 17, 'elements': [4, 5], 'cost': 20}, {'id': 18, 'elements': [4, 5, 8], 'cost': 9}, {'id': 19, 'elements': [6, 10], 'cost': 176}, {'id': 20, 'elements': [5, 6], 'cost': 42}, {'id': 21, 'elements': [2, 3], 'cost': 162}, {'id': 22, 'elements': [8, 10, 11, 12], 'cost': 100}, {'id': 23, 'elements': [6, 8, 9], 'cost': 141}, {'id': 24, 'elements': [11, 13], 'cost': 124}, {'id': 25, 'elements': [3, 4, 5, 6], 'cost': 204}, {'id': 26, 'elements': [5, 7, 8], 'cost': 15}, {'id': 27, 'elements': [4, 5, 6, 7], 'cost': 272}, {'id': 28, 'elements': [1, 3], 'cost': 110}, {'id': 29, 'elements': [5], 'cost': 73}, {'id': 30, 'elements': [3, 4, 5], 'cost': 204}, {'id': 31, 'elements': [8, 10], 'cost': 172}, {'id': 32, 'elements': [11], 'cost': 97}, {'id': 33, 'elements': [2], 'cost': 74}, {'id': 34, 'elements': [1, 2, 3, 4], 'cost': 216}, {'id': 35, 'elements': [7, 8, 9], 'cost': 177}, {'id': 36, 'elements': [9, 11, 13], 'cost': 240}, {'id': 37, 'elements': [1, 2], 'cost': 28}, {'id': 38, 'elements': [10, 11, 12, 13], 'cost': 192}, {'id': 39, 'elements': [10, 11, 12], 'cost': 102}, {'id': 40, 'elements': [1, 3, 4], 'cost': 159}, {'id': 41, 'elements': [4, 5, 6], 'cost': 102}, {'id': 42, 'elements': [10, 12, 13], 'cost': 255}, {'id': 43, 'elements': [2, 4, 5], 'cost': 84}, {'id': 44, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 45, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 46, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 47, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [13], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0029_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0029_bag_stats.png'}","[2, 11, 25, 37]",491.0,"{'num_elements': 13, 'num_sets': 56, 'sets': [{'id': 'S1', 'elements': ['A', 'B', 'E'], 'cost': 42}, {'id': 'S2', 'elements': ['K', 'L', 'M'], 'cost': 195}, {'id': 'S3', 'elements': ['A', 'B', 'C'], 'cost': 300}, {'id': 'S4', 'elements': ['E', 'H'], 'cost': 32}, {'id': 'S5', 'elements': ['L', 'M'], 'cost': 166}, {'id': 'S6', 'elements': ['D', 'F', 'G'], 'cost': 162}, {'id': 'S7', 'elements': ['F', 'G'], 'cost': 190}, {'id': 'S8', 'elements': ['E', 'F', 'G'], 'cost': 216}, {'id': 'S9', 'elements': ['B', 'C', 'D', 'F'], 'cost': 220}, {'id': 'S10', 'elements': ['K', 'L'], 'cost': 104}, {'id': 'S11', 'elements': ['G', 'H', 'I', 'J'], 'cost': 64}, {'id': 'S12', 'elements': ['I', 'L'], 'cost': 160}, {'id': 'S13', 'elements': ['A', 'B', 'D'], 'cost': 177}, {'id': 'S14', 'elements': ['I', 'J'], 'cost': 98}, {'id': 'S15', 'elements': ['I', 'J', 'K'], 'cost': 9}, {'id': 'S16', 'elements': ['H', 'K', 'L'], 'cost': 48}, {'id': 'S17', 'elements': ['D', 'E'], 'cost': 20}, {'id': 'S18', 'elements': ['D', 'E', 'H'], 'cost': 9}, {'id': 'S19', 'elements': ['F', 'J'], 'cost': 176}, {'id': 'S20', 'elements': ['E', 'F'], 'cost': 42}, {'id': 'S21', 'elements': ['B', 'C'], 'cost': 162}, {'id': 'S22', 'elements': ['H', 'J', 'K', 'L'], 'cost': 100}, {'id': 'S23', 'elements': ['F', 'H', 'I'], 'cost': 141}, {'id': 'S24', 'elements': ['K', 'M'], 'cost': 124}, {'id': 'S25', 'elements': ['C', 'D', 'E', 'F'], 'cost': 204}, {'id': 'S26', 'elements': ['E', 'G', 'H'], 'cost': 15}, {'id': 'S27', 'elements': ['D', 'E', 'F', 'G'], 'cost': 272}, {'id': 'S28', 'elements': ['A', 'C'], 'cost': 110}, {'id': 'S29', 'elements': ['E'], 'cost': 73}, {'id': 'S30', 'elements': ['C', 'D', 'E'], 'cost': 204}, {'id': 'S31', 'elements': ['H', 'J'], 'cost': 172}, {'id': 'S32', 'elements': ['K'], 'cost': 97}, {'id': 'S33', 'elements': ['B'], 'cost': 74}, {'id': 'S34', 'elements': ['A', 'B', 'C', 'D'], 'cost': 216}, {'id': 'S35', 'elements': ['G', 'H', 'I'], 'cost': 177}, {'id': 'S36', 'elements': ['I', 'K', 'M'], 'cost': 240}, {'id': 'S37', 'elements': ['A', 'B'], 'cost': 28}, {'id': 'S38', 'elements': ['J', 'K', 'L', 'M'], 'cost': 192}, {'id': 'S39', 'elements': ['J', 'K', 'L'], 'cost': 102}, {'id': 'S40', 'elements': ['A', 'C', 'D'], 'cost': 159}, {'id': 'S41', 'elements': ['D', 'E', 'F'], 'cost': 102}, {'id': 'S42', 'elements': ['J', 'L', 'M'], 'cost': 255}, {'id': 'S43', 'elements': ['B', 'D', 'E'], 'cost': 84}, {'id': 'S44', 'elements': ['A'], 'cost': 10000}, {'id': 'S45', 'elements': ['B'], 'cost': 10000}, {'id': 'S46', 'elements': ['C'], 'cost': 10000}, {'id': 'S47', 'elements': ['D'], 'cost': 10000}, {'id': 'S48', 'elements': ['E'], 'cost': 10000}, {'id': 'S49', 'elements': ['F'], 'cost': 10000}, {'id': 'S50', 'elements': ['G'], 'cost': 10000}, {'id': 'S51', 'elements': ['H'], 'cost': 10000}, {'id': 'S52', 'elements': ['I'], 'cost': 10000}, {'id': 'S53', 'elements': ['J'], 'cost': 10000}, {'id': 'S54', 'elements': ['K'], 'cost': 10000}, {'id': 'S55', 'elements': ['L'], 'cost': 10000}, {'id': 'S56', 'elements': ['M'], 'cost': 10000}]}","['S2', 'S11', 'S25', 'S37']",30,json,names
SPP,SPP,"I was helping a small club put together next season’s training schedule: there’s a list of athletes, and there’s a menu of training blocks that each address certain development needs. The job is to pick which blocks to run so that every athlete’s specific need is matched by one and only one block — nobody gets two different blocks and nobody is left uncovered — and then tally up the price tags for the blocks chosen; the plan with the lowest total cost is the one everyone prefers. The concrete list of athletes, blocks, and costs is shown below.

- **num_athletes**: 13
- **num_blocks**: 56

| block_id | block_cost | athletes_covered |
|---|---|---|
| S1 | 228 | 4 5 6 |
| S2 | 177 | 7 8 9 |
| S3 | 150 | 6 7 10 |
| S4 | 88 | 7 8 9 10 |
| S5 | 288 | 7 9 11 |
| S6 | 8 | 0 2 |
| S7 | 33 | 7 8 10 |
| S8 | 132 | 2 3 |
| S9 | 168 | 5 6 |
| S10 | 112 | 3 4 |
| S11 | 58 | 1 3 |
| S12 | 138 | 0 1 |
| S13 | 104 | 3 5 |
| S14 | 178 | 0 3 |
| S15 | 195 | 9 11 12 |
| S16 | 164 | 7 8 9 11 |
| S17 | 6 | 0 1 4 |
| S18 | 30 | 5 6 7 |
| S19 | 122 | 11 12 |
| S20 | 9 | 4 7 8 |
| S21 | 57 | 2 4 5 |
| S22 | 285 | 0 1 2 |
| S23 | 94 | 7 8 |
| S24 | 134 | 4 5 |
| S25 | 246 | 8 9 10 |
| S26 | 85 | 7 |
| S27 | 188 | 4 5 6 8 |
| S28 | 90 | 7 10 |
| S29 | 177 | 10 11 12 |
| S30 | 114 | 9 10 11 |
| S31 | 152 | 5 7 |
| S32 | 170 | 6 7 |
| S33 | 76 | 9 10 11 12 |
| S34 | 184 | 0 2 3 5 |
| S35 | 26 | 9 10 |
| S36 | 300 | 3 4 5 6 |
| S37 | 142 | 10 11 |
| S38 | 162 | 2 4 |
| S39 | 8 | 6 8 |
| S40 | 300 | 4 5 7 8 |
| S41 | 21 | 2 3 4 |
| S42 | 252 | 6 7 8 |
| S43 | 124 | 7 9 |
| S44 | 10000 | 0 |
| S45 | 10000 | 1 |
| S46 | 10000 | 2 |
| S47 | 10000 | 3 |
| S48 | 10000 | 4 |
| S49 | 10000 | 5 |
| S50 | 10000 | 6 |
| S51 | 10000 | 7 |
| S52 | 10000 | 8 |
| S53 | 10000 | 9 |
| S54 | 10000 | 10 |
| S55 | 10000 | 11 |
| S56 | 10000 | 12 |

Also, when you send the picks back, please use this simple JSON layout so it's easy to read by whatever tool is collecting the plans:

{
  ""solution"": [""block_id"", ...]
}

Think of that as a tiny form: ""solution"" holds an array of the training block IDs you choose (one string per block). The JSON above is just a sketch of the shape I expect — not the actual answer.

Please make sure to use the exact identifiers from the instance input — do not rename them or invent new ones. 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 13, 'num_sets': 56, 'density': 0.17582417582417584, 'sets': [{'id': 1, 'elements': [5, 6, 7], 'cost': 228}, {'id': 2, 'elements': [8, 9, 10], 'cost': 177}, {'id': 3, 'elements': [7, 8, 11], 'cost': 150}, {'id': 4, 'elements': [8, 9, 10, 11], 'cost': 88}, {'id': 5, 'elements': [8, 10, 12], 'cost': 288}, {'id': 6, 'elements': [1, 3], 'cost': 8}, {'id': 7, 'elements': [8, 9, 11], 'cost': 33}, {'id': 8, 'elements': [3, 4], 'cost': 132}, {'id': 9, 'elements': [6, 7], 'cost': 168}, {'id': 10, 'elements': [4, 5], 'cost': 112}, {'id': 11, 'elements': [2, 4], 'cost': 58}, {'id': 12, 'elements': [1, 2], 'cost': 138}, {'id': 13, 'elements': [4, 6], 'cost': 104}, {'id': 14, 'elements': [1, 4], 'cost': 178}, {'id': 15, 'elements': [10, 12, 13], 'cost': 195}, {'id': 16, 'elements': [8, 9, 10, 12], 'cost': 164}, {'id': 17, 'elements': [1, 2, 5], 'cost': 6}, {'id': 18, 'elements': [6, 7, 8], 'cost': 30}, {'id': 19, 'elements': [12, 13], 'cost': 122}, {'id': 20, 'elements': [5, 8, 9], 'cost': 9}, {'id': 21, 'elements': [3, 5, 6], 'cost': 57}, {'id': 22, 'elements': [1, 2, 3], 'cost': 285}, {'id': 23, 'elements': [8, 9], 'cost': 94}, {'id': 24, 'elements': [5, 6], 'cost': 134}, {'id': 25, 'elements': [9, 10, 11], 'cost': 246}, {'id': 26, 'elements': [8], 'cost': 85}, {'id': 27, 'elements': [5, 6, 7, 9], 'cost': 188}, {'id': 28, 'elements': [8, 11], 'cost': 90}, {'id': 29, 'elements': [11, 12, 13], 'cost': 177}, {'id': 30, 'elements': [10, 11, 12], 'cost': 114}, {'id': 31, 'elements': [6, 8], 'cost': 152}, {'id': 32, 'elements': [7, 8], 'cost': 170}, {'id': 33, 'elements': [10, 11, 12, 13], 'cost': 76}, {'id': 34, 'elements': [1, 3, 4, 6], 'cost': 184}, {'id': 35, 'elements': [10, 11], 'cost': 26}, {'id': 36, 'elements': [4, 5, 6, 7], 'cost': 300}, {'id': 37, 'elements': [11, 12], 'cost': 142}, {'id': 38, 'elements': [3, 5], 'cost': 162}, {'id': 39, 'elements': [7, 9], 'cost': 8}, {'id': 40, 'elements': [5, 6, 8, 9], 'cost': 300}, {'id': 41, 'elements': [3, 4, 5], 'cost': 21}, {'id': 42, 'elements': [7, 8, 9], 'cost': 252}, {'id': 43, 'elements': [8, 10], 'cost': 124}, {'id': 44, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 45, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 46, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 47, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [13], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0030_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0030_bag_stats.png'}","[6, 9, 11, 20, 33]",319.0,"{'num_elements': 13, 'num_sets': 56, 'sets': [{'id': 'S1', 'elements': [4, 5, 6], 'cost': 228}, {'id': 'S2', 'elements': [7, 8, 9], 'cost': 177}, {'id': 'S3', 'elements': [6, 7, 10], 'cost': 150}, {'id': 'S4', 'elements': [7, 8, 9, 10], 'cost': 88}, {'id': 'S5', 'elements': [7, 9, 11], 'cost': 288}, {'id': 'S6', 'elements': [0, 2], 'cost': 8}, {'id': 'S7', 'elements': [7, 8, 10], 'cost': 33}, {'id': 'S8', 'elements': [2, 3], 'cost': 132}, {'id': 'S9', 'elements': [5, 6], 'cost': 168}, {'id': 'S10', 'elements': [3, 4], 'cost': 112}, {'id': 'S11', 'elements': [1, 3], 'cost': 58}, {'id': 'S12', 'elements': [0, 1], 'cost': 138}, {'id': 'S13', 'elements': [3, 5], 'cost': 104}, {'id': 'S14', 'elements': [0, 3], 'cost': 178}, {'id': 'S15', 'elements': [9, 11, 12], 'cost': 195}, {'id': 'S16', 'elements': [7, 8, 9, 11], 'cost': 164}, {'id': 'S17', 'elements': [0, 1, 4], 'cost': 6}, {'id': 'S18', 'elements': [5, 6, 7], 'cost': 30}, {'id': 'S19', 'elements': [11, 12], 'cost': 122}, {'id': 'S20', 'elements': [4, 7, 8], 'cost': 9}, {'id': 'S21', 'elements': [2, 4, 5], 'cost': 57}, {'id': 'S22', 'elements': [0, 1, 2], 'cost': 285}, {'id': 'S23', 'elements': [7, 8], 'cost': 94}, {'id': 'S24', 'elements': [4, 5], 'cost': 134}, {'id': 'S25', 'elements': [8, 9, 10], 'cost': 246}, {'id': 'S26', 'elements': [7], 'cost': 85}, {'id': 'S27', 'elements': [4, 5, 6, 8], 'cost': 188}, {'id': 'S28', 'elements': [7, 10], 'cost': 90}, {'id': 'S29', 'elements': [10, 11, 12], 'cost': 177}, {'id': 'S30', 'elements': [9, 10, 11], 'cost': 114}, {'id': 'S31', 'elements': [5, 7], 'cost': 152}, {'id': 'S32', 'elements': [6, 7], 'cost': 170}, {'id': 'S33', 'elements': [9, 10, 11, 12], 'cost': 76}, {'id': 'S34', 'elements': [0, 2, 3, 5], 'cost': 184}, {'id': 'S35', 'elements': [9, 10], 'cost': 26}, {'id': 'S36', 'elements': [3, 4, 5, 6], 'cost': 300}, {'id': 'S37', 'elements': [10, 11], 'cost': 142}, {'id': 'S38', 'elements': [2, 4], 'cost': 162}, {'id': 'S39', 'elements': [6, 8], 'cost': 8}, {'id': 'S40', 'elements': [4, 5, 7, 8], 'cost': 300}, {'id': 'S41', 'elements': [2, 3, 4], 'cost': 21}, {'id': 'S42', 'elements': [6, 7, 8], 'cost': 252}, {'id': 'S43', 'elements': [7, 9], 'cost': 124}, {'id': 'S44', 'elements': [0], 'cost': 10000}, {'id': 'S45', 'elements': [1], 'cost': 10000}, {'id': 'S46', 'elements': [2], 'cost': 10000}, {'id': 'S47', 'elements': [3], 'cost': 10000}, {'id': 'S48', 'elements': [4], 'cost': 10000}, {'id': 'S49', 'elements': [5], 'cost': 10000}, {'id': 'S50', 'elements': [6], 'cost': 10000}, {'id': 'S51', 'elements': [7], 'cost': 10000}, {'id': 'S52', 'elements': [8], 'cost': 10000}, {'id': 'S53', 'elements': [9], 'cost': 10000}, {'id': 'S54', 'elements': [10], 'cost': 10000}, {'id': 'S55', 'elements': [11], 'cost': 10000}, {'id': 'S56', 'elements': [12], 'cost': 10000}]}","['S6', 'S9', 'S11', 'S20', 'S33']",31,markdown_table,0
SPP,SPP,"We’ve got a pile of artifacts and a bunch of pre-made exhibit bundles, and the job is to decide which bundles to run so each artifact appears in one and only one display (no misses, no duplicates), and the whole setup costs the least amount; the way to see how good a plan is is simply to add up the prices of the selected bundles. The exact bundle options and artifact assignments are listed below.

- **total_artifacts**: 18
- **total_bundles**: 66

| bundle_id | bundle_cost | artifact_ids |
|---|---|---|
| S1 | 261 | 3 5 8 |
| S2 | 252 | 1 4 5 9 |
| S3 | 52 | 1 5 6 8 |
| S4 | 24 | 3 4 6 |
| S5 | 395 | 9 10 11 12 14 |
| S6 | 340 | 2 4 5 6 |
| S7 | 114 | 15 16 17 |
| S8 | 153 | 4 5 7 |
| S9 | 111 | 2 4 6 |
| S10 | 255 | 9 12 14 |
| S11 | 236 | 1 2 4 5 |
| S12 | 135 | 5 6 8 |
| S13 | 220 | 14 15 16 17 18 |
| S14 | 300 | 14 15 16 18 |
| S15 | 90 | 10 11 14 |
| S16 | 84 | 4 5 |
| S17 | 45 | 6 7 9 10 12 |
| S18 | 20 | 7 11 |
| S19 | 126 | 12 14 15 |
| S20 | 148 | 14 18 |
| S21 | 84 | 5 6 7 8 |
| S22 | 243 | 15 17 18 |
| S23 | 350 | 2 3 4 6 7 |
| S24 | 201 | 11 13 15 |
| S25 | 9 | 1 4 5 |
| S26 | 27 | 9 11 13 |
| S27 | 325 | 3 6 7 8 9 |
| S28 | 102 | 2 3 5 |
| S29 | 72 | 1 2 3 5 |
| S30 | 93 | 12 13 14 |
| S31 | 90 | 11 12 14 |
| S32 | 52 | 1 4 |
| S33 | 396 | 9 12 13 14 |
| S34 | 171 | 3 4 5 |
| S35 | 16 | 11 12 |
| S36 | 207 | 2 4 5 |
| S37 | 228 | 1 2 3 4 |
| S38 | 64 | 2 3 |
| S39 | 350 | 2 3 4 5 6 |
| S40 | 117 | 11 13 14 |
| S41 | 44 | 1 2 3 6 |
| S42 | 360 | 2 3 6 8 |
| S43 | 105 | 11 16 18 |
| S44 | 237 | 10 12 16 |
| S45 | 81 | 1 2 3 |
| S46 | 294 | 10 14 15 |
| S47 | 102 | 1 3 4 |
| S48 | 470 | 5 6 7 8 9 |
| S49 | 10000 | 1 |
| S50 | 10000 | 2 |
| S51 | 10000 | 3 |
| S52 | 10000 | 4 |
| S53 | 10000 | 5 |
| S54 | 10000 | 6 |
| S55 | 10000 | 7 |
| S56 | 10000 | 8 |
| S57 | 10000 | 9 |
| S58 | 10000 | 10 |
| S59 | 10000 | 11 |
| S60 | 10000 | 12 |
| S61 | 10000 | 13 |
| S62 | 10000 | 14 |
| S63 | 10000 | 15 |
| S64 | 10000 | 16 |
| S65 | 10000 | 17 |
| S66 | 10000 | 18 |

If you want to hand me a candidate plan, just drop it in the little JSON shape below so I can read it programmatically — nothing fancy, just a list of the bundles you'd run.

{
  ""solution"": [""bundle_id"", ...]
}

""solution"" holds the list of bundles to run. Each ""bundle_id"" is a placeholder for one bundle label from the instance (so put the actual bundle labels there). Think of it like a short form: the file has one key, and that key points to a list of the bundle names you want to activate.

This JSON is just a sketch of the expected shape, not the final answer itself — replace the placeholder(s) with the exact bundle identifiers from the instance.

Please use the identifiers exactly as they appear in the instance input — no renaming and no new labels.  
For example: Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'problem_type': 'SPP', 'num_elements': 18, 'num_sets': 66, 'density': 0.15235690235690236, 'sets': [{'id': 1, 'elements': [3, 5, 8], 'cost': 261}, {'id': 2, 'elements': [1, 4, 5, 9], 'cost': 252}, {'id': 3, 'elements': [1, 5, 6, 8], 'cost': 52}, {'id': 4, 'elements': [3, 4, 6], 'cost': 24}, {'id': 5, 'elements': [9, 10, 11, 12, 14], 'cost': 395}, {'id': 6, 'elements': [2, 4, 5, 6], 'cost': 340}, {'id': 7, 'elements': [15, 16, 17], 'cost': 114}, {'id': 8, 'elements': [4, 5, 7], 'cost': 153}, {'id': 9, 'elements': [2, 4, 6], 'cost': 111}, {'id': 10, 'elements': [9, 12, 14], 'cost': 255}, {'id': 11, 'elements': [1, 2, 4, 5], 'cost': 236}, {'id': 12, 'elements': [5, 6, 8], 'cost': 135}, {'id': 13, 'elements': [14, 15, 16, 17, 18], 'cost': 220}, {'id': 14, 'elements': [14, 15, 16, 18], 'cost': 300}, {'id': 15, 'elements': [10, 11, 14], 'cost': 90}, {'id': 16, 'elements': [4, 5], 'cost': 84}, {'id': 17, 'elements': [6, 7, 9, 10, 12], 'cost': 45}, {'id': 18, 'elements': [7, 11], 'cost': 20}, {'id': 19, 'elements': [12, 14, 15], 'cost': 126}, {'id': 20, 'elements': [14, 18], 'cost': 148}, {'id': 21, 'elements': [5, 6, 7, 8], 'cost': 84}, {'id': 22, 'elements': [15, 17, 18], 'cost': 243}, {'id': 23, 'elements': [2, 3, 4, 6, 7], 'cost': 350}, {'id': 24, 'elements': [11, 13, 15], 'cost': 201}, {'id': 25, 'elements': [1, 4, 5], 'cost': 9}, {'id': 26, 'elements': [9, 11, 13], 'cost': 27}, {'id': 27, 'elements': [3, 6, 7, 8, 9], 'cost': 325}, {'id': 28, 'elements': [2, 3, 5], 'cost': 102}, {'id': 29, 'elements': [1, 2, 3, 5], 'cost': 72}, {'id': 30, 'elements': [12, 13, 14], 'cost': 93}, {'id': 31, 'elements': [11, 12, 14], 'cost': 90}, {'id': 32, 'elements': [1, 4], 'cost': 52}, {'id': 33, 'elements': [9, 12, 13, 14], 'cost': 396}, {'id': 34, 'elements': [3, 4, 5], 'cost': 171}, {'id': 35, 'elements': [11, 12], 'cost': 16}, {'id': 36, 'elements': [2, 4, 5], 'cost': 207}, {'id': 37, 'elements': [1, 2, 3, 4], 'cost': 228}, {'id': 38, 'elements': [2, 3], 'cost': 64}, {'id': 39, 'elements': [2, 3, 4, 5, 6], 'cost': 350}, {'id': 40, 'elements': [11, 13, 14], 'cost': 117}, {'id': 41, 'elements': [1, 2, 3, 6], 'cost': 44}, {'id': 42, 'elements': [2, 3, 6, 8], 'cost': 360}, {'id': 43, 'elements': [11, 16, 18], 'cost': 105}, {'id': 44, 'elements': [10, 12, 16], 'cost': 237}, {'id': 45, 'elements': [1, 2, 3], 'cost': 81}, {'id': 46, 'elements': [10, 14, 15], 'cost': 294}, {'id': 47, 'elements': [1, 3, 4], 'cost': 102}, {'id': 48, 'elements': [5, 6, 7, 8, 9], 'cost': 470}, {'id': 49, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [17], 'cost': 10000, 'singleton': True}, {'id': 66, 'elements': [18], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0031_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0031_bag_stats.png'}","[11, 22, 27, 40, 44]",1158.0,"{'num_elements': 18, 'num_sets': 66, 'sets': [{'id': 'S1', 'elements': [3, 5, 8], 'cost': 261}, {'id': 'S2', 'elements': [1, 4, 5, 9], 'cost': 252}, {'id': 'S3', 'elements': [1, 5, 6, 8], 'cost': 52}, {'id': 'S4', 'elements': [3, 4, 6], 'cost': 24}, {'id': 'S5', 'elements': [9, 10, 11, 12, 14], 'cost': 395}, {'id': 'S6', 'elements': [2, 4, 5, 6], 'cost': 340}, {'id': 'S7', 'elements': [15, 16, 17], 'cost': 114}, {'id': 'S8', 'elements': [4, 5, 7], 'cost': 153}, {'id': 'S9', 'elements': [2, 4, 6], 'cost': 111}, {'id': 'S10', 'elements': [9, 12, 14], 'cost': 255}, {'id': 'S11', 'elements': [1, 2, 4, 5], 'cost': 236}, {'id': 'S12', 'elements': [5, 6, 8], 'cost': 135}, {'id': 'S13', 'elements': [14, 15, 16, 17, 18], 'cost': 220}, {'id': 'S14', 'elements': [14, 15, 16, 18], 'cost': 300}, {'id': 'S15', 'elements': [10, 11, 14], 'cost': 90}, {'id': 'S16', 'elements': [4, 5], 'cost': 84}, {'id': 'S17', 'elements': [6, 7, 9, 10, 12], 'cost': 45}, {'id': 'S18', 'elements': [7, 11], 'cost': 20}, {'id': 'S19', 'elements': [12, 14, 15], 'cost': 126}, {'id': 'S20', 'elements': [14, 18], 'cost': 148}, {'id': 'S21', 'elements': [5, 6, 7, 8], 'cost': 84}, {'id': 'S22', 'elements': [15, 17, 18], 'cost': 243}, {'id': 'S23', 'elements': [2, 3, 4, 6, 7], 'cost': 350}, {'id': 'S24', 'elements': [11, 13, 15], 'cost': 201}, {'id': 'S25', 'elements': [1, 4, 5], 'cost': 9}, {'id': 'S26', 'elements': [9, 11, 13], 'cost': 27}, {'id': 'S27', 'elements': [3, 6, 7, 8, 9], 'cost': 325}, {'id': 'S28', 'elements': [2, 3, 5], 'cost': 102}, {'id': 'S29', 'elements': [1, 2, 3, 5], 'cost': 72}, {'id': 'S30', 'elements': [12, 13, 14], 'cost': 93}, {'id': 'S31', 'elements': [11, 12, 14], 'cost': 90}, {'id': 'S32', 'elements': [1, 4], 'cost': 52}, {'id': 'S33', 'elements': [9, 12, 13, 14], 'cost': 396}, {'id': 'S34', 'elements': [3, 4, 5], 'cost': 171}, {'id': 'S35', 'elements': [11, 12], 'cost': 16}, {'id': 'S36', 'elements': [2, 4, 5], 'cost': 207}, {'id': 'S37', 'elements': [1, 2, 3, 4], 'cost': 228}, {'id': 'S38', 'elements': [2, 3], 'cost': 64}, {'id': 'S39', 'elements': [2, 3, 4, 5, 6], 'cost': 350}, {'id': 'S40', 'elements': [11, 13, 14], 'cost': 117}, {'id': 'S41', 'elements': [1, 2, 3, 6], 'cost': 44}, {'id': 'S42', 'elements': [2, 3, 6, 8], 'cost': 360}, {'id': 'S43', 'elements': [11, 16, 18], 'cost': 105}, {'id': 'S44', 'elements': [10, 12, 16], 'cost': 237}, {'id': 'S45', 'elements': [1, 2, 3], 'cost': 81}, {'id': 'S46', 'elements': [10, 14, 15], 'cost': 294}, {'id': 'S47', 'elements': [1, 3, 4], 'cost': 102}, {'id': 'S48', 'elements': [5, 6, 7, 8, 9], 'cost': 470}, {'id': 'S49', 'elements': [1], 'cost': 10000}, {'id': 'S50', 'elements': [2], 'cost': 10000}, {'id': 'S51', 'elements': [3], 'cost': 10000}, {'id': 'S52', 'elements': [4], 'cost': 10000}, {'id': 'S53', 'elements': [5], 'cost': 10000}, {'id': 'S54', 'elements': [6], 'cost': 10000}, {'id': 'S55', 'elements': [7], 'cost': 10000}, {'id': 'S56', 'elements': [8], 'cost': 10000}, {'id': 'S57', 'elements': [9], 'cost': 10000}, {'id': 'S58', 'elements': [10], 'cost': 10000}, {'id': 'S59', 'elements': [11], 'cost': 10000}, {'id': 'S60', 'elements': [12], 'cost': 10000}, {'id': 'S61', 'elements': [13], 'cost': 10000}, {'id': 'S62', 'elements': [14], 'cost': 10000}, {'id': 'S63', 'elements': [15], 'cost': 10000}, {'id': 'S64', 'elements': [16], 'cost': 10000}, {'id': 'S65', 'elements': [17], 'cost': 10000}, {'id': 'S66', 'elements': [18], 'cost': 10000}]}","['S11', 'S22', 'S27', 'S40', 'S44']",32,markdown_table,1
SPP,SPP,"Recently a couple looked over a bunch of vendor packages, and each package takes care of different pieces of the day’s logistics. They must pick some packages so every single task is covered exactly once (no missing services and no two packages doing the same job), and they want the cheapest overall bill — the cost is simply the sum of the chosen package prices. The concrete list of packages and tasks follows below.

There are 18 distinct wedding tasks they must cover exactly once, and 86 vendor bundles available to choose from.
Vendor bundle S1 handles tasks 14 15 16 17 18 and costs 300.
Vendor bundle S2 handles tasks 15 16 18 and costs 93.
Vendor bundle S3 handles tasks 1 3 and costs 72.
Vendor bundle S4 handles tasks 7 9 and costs 148.
Vendor bundle S5 handles tasks 8 10 11 and costs 297.
Vendor bundle S6 handles tasks 3 5 6 7 and costs 228.
Vendor bundle S7 handles tasks 11 12 15 and costs 246.
Vendor bundle S8 handles tasks 4 6 7 8 10 and costs 45.
Vendor bundle S9 handles tasks 4 7 and costs 14.
Vendor bundle S10 handles tasks 1 2 3 and costs 123.
Vendor bundle S11 handles tasks 5 9 and costs 68.
Vendor bundle S12 handles tasks 1 2 3 4 5 and costs 225.
Vendor bundle S13 handles tasks 16 17 and costs 62.
Vendor bundle S14 handles tasks 9 12 14 and costs 165.
Vendor bundle S15 handles tasks 2 4 6 8 9 and costs 240.
Vendor bundle S16 handles tasks 7 8 11 and costs 60.
Vendor bundle S17 handles tasks 9 11 12 13 15 and costs 490.
Vendor bundle S18 handles tasks 3 5 6 7 8 and costs 405.
Vendor bundle S19 handles tasks 12 14 15 16 and costs 312.
Vendor bundle S20 handles tasks 15 16 17 18 and costs 128.
Vendor bundle S21 handles tasks 11 12 16 and costs 267.
Vendor bundle S22 handles tasks 13 14 15 and costs 141.
Vendor bundle S23 handles tasks 7 8 9 11 and costs 68.
Vendor bundle S24 handles tasks 5 6 7 8 9 and costs 185.
Vendor bundle S25 handles tasks 2 4 8 and costs 222.
Vendor bundle S26 handles tasks 3 5 8 9 and costs 136.
Vendor bundle S27 handles tasks 12 14 16 17 18 and costs 50.
Vendor bundle S28 handles tasks 17 18 and costs 92.
Vendor bundle S29 handles tasks 14 15 and costs 162.
Vendor bundle S30 handles tasks 11 13 14 and costs 72.
Vendor bundle S31 handles tasks 14 15 16 18 and costs 336.
Vendor bundle S32 handles tasks 10 11 12 and costs 297.
Vendor bundle S33 handles tasks 7 10 and costs 134.
Vendor bundle S34 handles tasks 13 15 16 17 18 and costs 95.
Vendor bundle S35 handles tasks 6 7 and costs 158.
Vendor bundle S36 handles tasks 5 9 10 and costs 291.
Vendor bundle S37 handles tasks 6 9 and costs 54.
Vendor bundle S38 handles tasks 13 16 17 18 and costs 280.
Vendor bundle S39 handles tasks 3 4 6 9 and costs 296.
Vendor bundle S40 handles tasks 13 15 17 and costs 45.
Vendor bundle S41 handles tasks 2 3 4 7 and costs 144.
Vendor bundle S42 handles tasks 2 4 5 7 and costs 40.
Vendor bundle S43 handles tasks 16 17 18 and costs 261.
Vendor bundle S44 handles tasks 8 9 11 12 13 and costs 125.
Vendor bundle S45 handles tasks 1 2 3 5 6 and costs 145.
Vendor bundle S46 handles tasks 7 8 9 10 and costs 68.
Vendor bundle S47 handles tasks 4 5 6 and costs 3.
Vendor bundle S48 handles tasks 14 15 16 and costs 162.
Vendor bundle S49 handles tasks 11 12 13 14 and costs 96.
Vendor bundle S50 handles tasks 6 7 8 9 10 and costs 350.
Vendor bundle S51 handles tasks 12 14 16 and costs 180.
Vendor bundle S52 handles tasks 2 3 4 6 7 and costs 315.
Vendor bundle S53 handles tasks 5 7 8 11 and costs 212.
Vendor bundle S54 handles tasks 10 11 and costs 102.
Vendor bundle S55 handles tasks 12 13 14 15 17 and costs 30.
Vendor bundle S56 handles tasks 13 14 16 and costs 300.
Vendor bundle S57 handles tasks 7 9 11 14 and costs 196.
Vendor bundle S58 handles tasks 12 13 and costs 142.
Vendor bundle S59 handles tasks 2 3 and costs 52.
Vendor bundle S60 handles tasks 14 17 18 and costs 9.
Vendor bundle S61 handles tasks 11 12 14 and costs 33.
Vendor bundle S62 handles tasks 2 3 5 and costs 39.
Vendor bundle S63 handles tasks 1 2 3 6 and costs 296.
Vendor bundle S64 handles tasks 8 9 10 and costs 108.
Vendor bundle S65 handles tasks 14 16 17 and costs 243.
Vendor bundle S66 handles tasks 1 2 4 5 7 and costs 260.
Vendor bundle S67 handles tasks 12 16 17 18 and costs 256.
Vendor bundle S68 handles tasks 7 11 12 and costs 267.
Vendor bundle S69 handles tasks 1 and costs 10000.
Vendor bundle S70 handles tasks 2 and costs 10000.
Vendor bundle S71 handles tasks 3 and costs 10000.
Vendor bundle S72 handles tasks 4 and costs 10000.
Vendor bundle S73 handles tasks 5 and costs 10000.
Vendor bundle S74 handles tasks 6 and costs 10000.
Vendor bundle S75 handles tasks 7 and costs 10000.
Vendor bundle S76 handles tasks 8 and costs 10000.
Vendor bundle S77 handles tasks 9 and costs 10000.
Vendor bundle S78 handles tasks 10 and costs 10000.
Vendor bundle S79 handles tasks 11 and costs 10000.
Vendor bundle S80 handles tasks 12 and costs 10000.
Vendor bundle S81 handles tasks 13 and costs 10000.
Vendor bundle S82 handles tasks 14 and costs 10000.
Vendor bundle S83 handles tasks 15 and costs 10000.
Vendor bundle S84 handles tasks 16 and costs 10000.
Vendor bundle S85 handles tasks 17 and costs 10000.
Vendor bundle S86 handles tasks 18 and costs 10000.
They will select some of these bundles so every one of the 18 tasks is covered exactly once at the lowest total cost.

Oh, and when you send your pick, please stick to this simple JSON shape so it's easy to read and check:

{
  ""solution"": [""package_id"", ...]
}

Here ""solution"" is the list of packages you choose for the day — each entry should be the exact package identifier from the instance (one per chosen package). This JSON is just a sketch of the expected shape, not the final answer itself.

Please use the identifiers exactly as they appear in the instance input — do not rename them or invent new labels.
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 18, 'num_sets': 86, 'density': 0.16472868217054262, 'sets': [{'id': 1, 'elements': [14, 15, 16, 17, 18], 'cost': 300}, {'id': 2, 'elements': [15, 16, 18], 'cost': 93}, {'id': 3, 'elements': [1, 3], 'cost': 72}, {'id': 4, 'elements': [7, 9], 'cost': 148}, {'id': 5, 'elements': [8, 10, 11], 'cost': 297}, {'id': 6, 'elements': [3, 5, 6, 7], 'cost': 228}, {'id': 7, 'elements': [11, 12, 15], 'cost': 246}, {'id': 8, 'elements': [4, 6, 7, 8, 10], 'cost': 45}, {'id': 9, 'elements': [4, 7], 'cost': 14}, {'id': 10, 'elements': [1, 2, 3], 'cost': 123}, {'id': 11, 'elements': [5, 9], 'cost': 68}, {'id': 12, 'elements': [1, 2, 3, 4, 5], 'cost': 225}, {'id': 13, 'elements': [16, 17], 'cost': 62}, {'id': 14, 'elements': [9, 12, 14], 'cost': 165}, {'id': 15, 'elements': [2, 4, 6, 8, 9], 'cost': 240}, {'id': 16, 'elements': [7, 8, 11], 'cost': 60}, {'id': 17, 'elements': [9, 11, 12, 13, 15], 'cost': 490}, {'id': 18, 'elements': [3, 5, 6, 7, 8], 'cost': 405}, {'id': 19, 'elements': [12, 14, 15, 16], 'cost': 312}, {'id': 20, 'elements': [15, 16, 17, 18], 'cost': 128}, {'id': 21, 'elements': [11, 12, 16], 'cost': 267}, {'id': 22, 'elements': [13, 14, 15], 'cost': 141}, {'id': 23, 'elements': [7, 8, 9, 11], 'cost': 68}, {'id': 24, 'elements': [5, 6, 7, 8, 9], 'cost': 185}, {'id': 25, 'elements': [2, 4, 8], 'cost': 222}, {'id': 26, 'elements': [3, 5, 8, 9], 'cost': 136}, {'id': 27, 'elements': [12, 14, 16, 17, 18], 'cost': 50}, {'id': 28, 'elements': [17, 18], 'cost': 92}, {'id': 29, 'elements': [14, 15], 'cost': 162}, {'id': 30, 'elements': [11, 13, 14], 'cost': 72}, {'id': 31, 'elements': [14, 15, 16, 18], 'cost': 336}, {'id': 32, 'elements': [10, 11, 12], 'cost': 297}, {'id': 33, 'elements': [7, 10], 'cost': 134}, {'id': 34, 'elements': [13, 15, 16, 17, 18], 'cost': 95}, {'id': 35, 'elements': [6, 7], 'cost': 158}, {'id': 36, 'elements': [5, 9, 10], 'cost': 291}, {'id': 37, 'elements': [6, 9], 'cost': 54}, {'id': 38, 'elements': [13, 16, 17, 18], 'cost': 280}, {'id': 39, 'elements': [3, 4, 6, 9], 'cost': 296}, {'id': 40, 'elements': [13, 15, 17], 'cost': 45}, {'id': 41, 'elements': [2, 3, 4, 7], 'cost': 144}, {'id': 42, 'elements': [2, 4, 5, 7], 'cost': 40}, {'id': 43, 'elements': [16, 17, 18], 'cost': 261}, {'id': 44, 'elements': [8, 9, 11, 12, 13], 'cost': 125}, {'id': 45, 'elements': [1, 2, 3, 5, 6], 'cost': 145}, {'id': 46, 'elements': [7, 8, 9, 10], 'cost': 68}, {'id': 47, 'elements': [4, 5, 6], 'cost': 3}, {'id': 48, 'elements': [14, 15, 16], 'cost': 162}, {'id': 49, 'elements': [11, 12, 13, 14], 'cost': 96}, {'id': 50, 'elements': [6, 7, 8, 9, 10], 'cost': 350}, {'id': 51, 'elements': [12, 14, 16], 'cost': 180}, {'id': 52, 'elements': [2, 3, 4, 6, 7], 'cost': 315}, {'id': 53, 'elements': [5, 7, 8, 11], 'cost': 212}, {'id': 54, 'elements': [10, 11], 'cost': 102}, {'id': 55, 'elements': [12, 13, 14, 15, 17], 'cost': 30}, {'id': 56, 'elements': [13, 14, 16], 'cost': 300}, {'id': 57, 'elements': [7, 9, 11, 14], 'cost': 196}, {'id': 58, 'elements': [12, 13], 'cost': 142}, {'id': 59, 'elements': [2, 3], 'cost': 52}, {'id': 60, 'elements': [14, 17, 18], 'cost': 9}, {'id': 61, 'elements': [11, 12, 14], 'cost': 33}, {'id': 62, 'elements': [2, 3, 5], 'cost': 39}, {'id': 63, 'elements': [1, 2, 3, 6], 'cost': 296}, {'id': 64, 'elements': [8, 9, 10], 'cost': 108}, {'id': 65, 'elements': [14, 16, 17], 'cost': 243}, {'id': 66, 'elements': [1, 2, 4, 5, 7], 'cost': 260}, {'id': 67, 'elements': [12, 16, 17, 18], 'cost': 256}, {'id': 68, 'elements': [7, 11, 12], 'cost': 267}, {'id': 69, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 70, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 71, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 72, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 73, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 74, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 75, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 76, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 77, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 78, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 79, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 80, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 81, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 82, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 83, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 84, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 85, 'elements': [17], 'cost': 10000, 'singleton': True}, {'id': 86, 'elements': [18], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0032_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0032_bag_stats.png'}","[10, 34, 46, 47, 61]",322.0,"{'num_elements': 18, 'num_sets': 86, 'sets': [{'id': 'S1', 'elements': [14, 15, 16, 17, 18], 'cost': 300}, {'id': 'S2', 'elements': [15, 16, 18], 'cost': 93}, {'id': 'S3', 'elements': [1, 3], 'cost': 72}, {'id': 'S4', 'elements': [7, 9], 'cost': 148}, {'id': 'S5', 'elements': [8, 10, 11], 'cost': 297}, {'id': 'S6', 'elements': [3, 5, 6, 7], 'cost': 228}, {'id': 'S7', 'elements': [11, 12, 15], 'cost': 246}, {'id': 'S8', 'elements': [4, 6, 7, 8, 10], 'cost': 45}, {'id': 'S9', 'elements': [4, 7], 'cost': 14}, {'id': 'S10', 'elements': [1, 2, 3], 'cost': 123}, {'id': 'S11', 'elements': [5, 9], 'cost': 68}, {'id': 'S12', 'elements': [1, 2, 3, 4, 5], 'cost': 225}, {'id': 'S13', 'elements': [16, 17], 'cost': 62}, {'id': 'S14', 'elements': [9, 12, 14], 'cost': 165}, {'id': 'S15', 'elements': [2, 4, 6, 8, 9], 'cost': 240}, {'id': 'S16', 'elements': [7, 8, 11], 'cost': 60}, {'id': 'S17', 'elements': [9, 11, 12, 13, 15], 'cost': 490}, {'id': 'S18', 'elements': [3, 5, 6, 7, 8], 'cost': 405}, {'id': 'S19', 'elements': [12, 14, 15, 16], 'cost': 312}, {'id': 'S20', 'elements': [15, 16, 17, 18], 'cost': 128}, {'id': 'S21', 'elements': [11, 12, 16], 'cost': 267}, {'id': 'S22', 'elements': [13, 14, 15], 'cost': 141}, {'id': 'S23', 'elements': [7, 8, 9, 11], 'cost': 68}, {'id': 'S24', 'elements': [5, 6, 7, 8, 9], 'cost': 185}, {'id': 'S25', 'elements': [2, 4, 8], 'cost': 222}, {'id': 'S26', 'elements': [3, 5, 8, 9], 'cost': 136}, {'id': 'S27', 'elements': [12, 14, 16, 17, 18], 'cost': 50}, {'id': 'S28', 'elements': [17, 18], 'cost': 92}, {'id': 'S29', 'elements': [14, 15], 'cost': 162}, {'id': 'S30', 'elements': [11, 13, 14], 'cost': 72}, {'id': 'S31', 'elements': [14, 15, 16, 18], 'cost': 336}, {'id': 'S32', 'elements': [10, 11, 12], 'cost': 297}, {'id': 'S33', 'elements': [7, 10], 'cost': 134}, {'id': 'S34', 'elements': [13, 15, 16, 17, 18], 'cost': 95}, {'id': 'S35', 'elements': [6, 7], 'cost': 158}, {'id': 'S36', 'elements': [5, 9, 10], 'cost': 291}, {'id': 'S37', 'elements': [6, 9], 'cost': 54}, {'id': 'S38', 'elements': [13, 16, 17, 18], 'cost': 280}, {'id': 'S39', 'elements': [3, 4, 6, 9], 'cost': 296}, {'id': 'S40', 'elements': [13, 15, 17], 'cost': 45}, {'id': 'S41', 'elements': [2, 3, 4, 7], 'cost': 144}, {'id': 'S42', 'elements': [2, 4, 5, 7], 'cost': 40}, {'id': 'S43', 'elements': [16, 17, 18], 'cost': 261}, {'id': 'S44', 'elements': [8, 9, 11, 12, 13], 'cost': 125}, {'id': 'S45', 'elements': [1, 2, 3, 5, 6], 'cost': 145}, {'id': 'S46', 'elements': [7, 8, 9, 10], 'cost': 68}, {'id': 'S47', 'elements': [4, 5, 6], 'cost': 3}, {'id': 'S48', 'elements': [14, 15, 16], 'cost': 162}, {'id': 'S49', 'elements': [11, 12, 13, 14], 'cost': 96}, {'id': 'S50', 'elements': [6, 7, 8, 9, 10], 'cost': 350}, {'id': 'S51', 'elements': [12, 14, 16], 'cost': 180}, {'id': 'S52', 'elements': [2, 3, 4, 6, 7], 'cost': 315}, {'id': 'S53', 'elements': [5, 7, 8, 11], 'cost': 212}, {'id': 'S54', 'elements': [10, 11], 'cost': 102}, {'id': 'S55', 'elements': [12, 13, 14, 15, 17], 'cost': 30}, {'id': 'S56', 'elements': [13, 14, 16], 'cost': 300}, {'id': 'S57', 'elements': [7, 9, 11, 14], 'cost': 196}, {'id': 'S58', 'elements': [12, 13], 'cost': 142}, {'id': 'S59', 'elements': [2, 3], 'cost': 52}, {'id': 'S60', 'elements': [14, 17, 18], 'cost': 9}, {'id': 'S61', 'elements': [11, 12, 14], 'cost': 33}, {'id': 'S62', 'elements': [2, 3, 5], 'cost': 39}, {'id': 'S63', 'elements': [1, 2, 3, 6], 'cost': 296}, {'id': 'S64', 'elements': [8, 9, 10], 'cost': 108}, {'id': 'S65', 'elements': [14, 16, 17], 'cost': 243}, {'id': 'S66', 'elements': [1, 2, 4, 5, 7], 'cost': 260}, {'id': 'S67', 'elements': [12, 16, 17, 18], 'cost': 256}, {'id': 'S68', 'elements': [7, 11, 12], 'cost': 267}, {'id': 'S69', 'elements': [1], 'cost': 10000}, {'id': 'S70', 'elements': [2], 'cost': 10000}, {'id': 'S71', 'elements': [3], 'cost': 10000}, {'id': 'S72', 'elements': [4], 'cost': 10000}, {'id': 'S73', 'elements': [5], 'cost': 10000}, {'id': 'S74', 'elements': [6], 'cost': 10000}, {'id': 'S75', 'elements': [7], 'cost': 10000}, {'id': 'S76', 'elements': [8], 'cost': 10000}, {'id': 'S77', 'elements': [9], 'cost': 10000}, {'id': 'S78', 'elements': [10], 'cost': 10000}, {'id': 'S79', 'elements': [11], 'cost': 10000}, {'id': 'S80', 'elements': [12], 'cost': 10000}, {'id': 'S81', 'elements': [13], 'cost': 10000}, {'id': 'S82', 'elements': [14], 'cost': 10000}, {'id': 'S83', 'elements': [15], 'cost': 10000}, {'id': 'S84', 'elements': [16], 'cost': 10000}, {'id': 'S85', 'elements': [17], 'cost': 10000}, {'id': 'S86', 'elements': [18], 'cost': 10000}]}","['S10', 'S34', 'S46', 'S47', 'S61']",33,nl,1
SPP,SPP,"Imagine the HR team must pick which benefit packages to use and slot every person into just one package — nobody can be left uncovered or put into two packages. Each package has a price tag for the employer, and the goal is to make the company’s total payout as small as possible by adding up the employer contributions of the packages that are selected. The concrete options and numbers are shown below.

{
  ""total_employees"": 17,
  ""num_bundles"": 76,
  ""sets"": [
    {
      ""bundle_id"": ""S1"",
      ""covered_employee_ids"": [
        7,
        8,
        9,
        10,
        11
      ],
      ""employer_contribution"": 420
    },
    {
      ""bundle_id"": ""S2"",
      ""covered_employee_ids"": [
        14,
        16,
        17
      ],
      ""employer_contribution"": 282
    },
    {
      ""bundle_id"": ""S3"",
      ""covered_employee_ids"": [
        2,
        4
      ],
      ""employer_contribution"": 34
    },
    {
      ""bundle_id"": ""S4"",
      ""covered_employee_ids"": [
        4,
        5,
        6,
        9
      ],
      ""employer_contribution"": 60
    },
    {
      ""bundle_id"": ""S5"",
      ""covered_employee_ids"": [
        14,
        15
      ],
      ""employer_contribution"": 136
    },
    {
      ""bundle_id"": ""S6"",
      ""covered_employee_ids"": [
        13,
        14,
        15
      ],
      ""employer_contribution"": 114
    },
    {
      ""bundle_id"": ""S7"",
      ""covered_employee_ids"": [
        8,
        9,
        10,
        11
      ],
      ""employer_contribution"": 76
    },
    {
      ""bundle_id"": ""S8"",
      ""covered_employee_ids"": [
        6,
        8,
        10
      ],
      ""employer_contribution"": 234
    },
    {
      ""bundle_id"": ""S9"",
      ""covered_employee_ids"": [
        14,
        15,
        16
      ],
      ""employer_contribution"": 54
    },
    {
      ""bundle_id"": ""S10"",
      ""covered_employee_ids"": [
        12,
        15
      ],
      ""employer_contribution"": 40
    },
    {
      ""bundle_id"": ""S11"",
      ""covered_employee_ids"": [
        1,
        2,
        4
      ],
      ""employer_contribution"": 207
    },
    {
      ""bundle_id"": ""S12"",
      ""covered_employee_ids"": [
        2,
        3,
        4
      ],
      ""employer_contribution"": 135
    },
    {
      ""bundle_id"": ""S13"",
      ""covered_employee_ids"": [
        4,
        6,
        7,
        8
      ],
      ""employer_contribution"": 220
    },
    {
      ""bundle_id"": ""S14"",
      ""covered_employee_ids"": [
        9,
        10,
        11,
        13
      ],
      ""employer_contribution"": 332
    },
    {
      ""bundle_id"": ""S15"",
      ""covered_employee_ids"": [
        1,
        2,
        3,
        4
      ],
      ""employer_contribution"": 292
    },
    {
      ""bundle_id"": ""S16"",
      ""covered_employee_ids"": [
        16,
        17
      ],
      ""employer_contribution"": 86
    },
    {
      ""bundle_id"": ""S17"",
      ""covered_employee_ids"": [
        9,
        10,
        11,
        12
      ],
      ""employer_contribution"": 108
    },
    {
      ""bundle_id"": ""S18"",
      ""covered_employee_ids"": [
        6,
        12
      ],
      ""employer_contribution"": 158
    },
    {
      ""bundle_id"": ""S19"",
      ""covered_employee_ids"": [
        1,
        6
      ],
      ""employer_contribution"": 20
    },
    {
      ""bundle_id"": ""S20"",
      ""covered_employee_ids"": [
        1,
        2,
        3,
        4,
        5
      ],
      ""employer_contribution"": 490
    },
    {
      ""bundle_id"": ""S21"",
      ""covered_employee_ids"": [
        12,
        13
      ],
      ""employer_contribution"": 122
    },
    {
      ""bundle_id"": ""S22"",
      ""covered_employee_ids"": [
        7,
        9
      ],
      ""employer_contribution"": 184
    },
    {
      ""bundle_id"": ""S23"",
      ""covered_employee_ids"": [
        13,
        14
      ],
      ""employer_contribution"": 66
    },
    {
      ""bundle_id"": ""S24"",
      ""covered_employee_ids"": [
        12,
        14,
        15,
        16
      ],
      ""employer_contribution"": 396
    },
    {
      ""bundle_id"": ""S25"",
      ""covered_employee_ids"": [
        13,
        15,
        16,
        17
      ],
      ""employer_contribution"": 196
    },
    {
      ""bundle_id"": ""S26"",
      ""covered_employee_ids"": [
        1,
        2,
        3
      ],
      ""employer_contribution"": 228
    },
    {
      ""bundle_id"": ""S27"",
      ""covered_employee_ids"": [
        3,
        5,
        6
      ],
      ""employer_contribution"": 6
    },
    {
      ""bundle_id"": ""S28"",
      ""covered_employee_ids"": [
        1,
        2,
        3,
        7
      ],
      ""employer_contribution"": 364
    },
    {
      ""bundle_id"": ""S29"",
      ""covered_employee_ids"": [
        3,
        4,
        5
      ],
      ""employer_contribution"": 183
    },
    {
      ""bundle_id"": ""S30"",
      ""covered_employee_ids"": [
        13,
        14,
        15,
        16,
        17
      ],
      ""employer_contribution"": 25
    },
    {
      ""bundle_id"": ""S31"",
      ""covered_employee_ids"": [
        3,
        6,
        7
      ],
      ""employer_contribution"": 33
    },
    {
      ""bundle_id"": ""S32"",
      ""covered_employee_ids"": [
        1,
        2,
        5
      ],
      ""employer_contribution"": 39
    },
    {
      ""bundle_id"": ""S33"",
      ""covered_employee_ids"": [
        5,
        8,
        10,
        11,
        13
      ],
      ""employer_contribution"": 205
    },
    {
      ""bundle_id"": ""S34"",
      ""covered_employee_ids"": [
        5,
        6,
        7,
        8,
        9
      ],
      ""employer_contribution"": 260
    },
    {
      ""bundle_id"": ""S35"",
      ""covered_employee_ids"": [
        1,
        3,
        4
      ],
      ""employer_contribution"": 282
    },
    {
      ""bundle_id"": ""S36"",
      ""covered_employee_ids"": [
        4,
        5,
        6
      ],
      ""employer_contribution"": 195
    },
    {
      ""bundle_id"": ""S37"",
      ""covered_employee_ids"": [
        3,
        4,
        5,
        7,
        8
      ],
      ""employer_contribution"": 160
    },
    {
      ""bundle_id"": ""S38"",
      ""covered_employee_ids"": [
        5,
        7,
        8
      ],
      ""employer_contribution"": 147
    },
    {
      ""bundle_id"": ""S39"",
      ""covered_employee_ids"": [
        5,
        6,
        7,
        9
      ],
      ""employer_contribution"": 140
    },
    {
      ""bundle_id"": ""S40"",
      ""covered_employee_ids"": [
        3,
        5,
        6,
        7,
        9
      ],
      ""employer_contribution"": 235
    },
    {
      ""bundle_id"": ""S41"",
      ""covered_employee_ids"": [
        15,
        16,
        17
      ],
      ""employer_contribution"": 15
    },
    {
      ""bundle_id"": ""S42"",
      ""covered_employee_ids"": [
        11,
        13,
        14,
        15,
        17
      ],
      ""employer_contribution"": 440
    },
    {
      ""bundle_id"": ""S43"",
      ""covered_employee_ids"": [
        6,
        7,
        9,
        10
      ],
      ""employer_contribution"": 140
    },
    {
      ""bundle_id"": ""S44"",
      ""covered_employee_ids"": [
        6,
        7
      ],
      ""employer_contribution"": 10
    },
    {
      ""bundle_id"": ""S45"",
      ""covered_employee_ids"": [
        12,
        13,
        15,
        16
      ],
      ""employer_contribution"": 308
    },
    {
      ""bundle_id"": ""S46"",
      ""covered_employee_ids"": [
        12,
        15,
        16
      ],
      ""employer_contribution"": 294
    },
    {
      ""bundle_id"": ""S47"",
      ""covered_employee_ids"": [
        7,
        8
      ],
      ""employer_contribution"": 180
    },
    {
      ""bundle_id"": ""S48"",
      ""covered_employee_ids"": [
        5,
        7,
        8,
        9,
        10
      ],
      ""employer_contribution"": 35
    },
    {
      ""bundle_id"": ""S49"",
      ""covered_employee_ids"": [
        8,
        9,
        10,
        11,
        12
      ],
      ""employer_contribution"": 425
    },
    {
      ""bundle_id"": ""S50"",
      ""covered_employee_ids"": [
        3,
        5
      ],
      ""employer_contribution"": 62
    },
    {
      ""bundle_id"": ""S51"",
      ""covered_employee_ids"": [
        12,
        13,
        14,
        15,
        16
      ],
      ""employer_contribution"": 310
    },
    {
      ""bundle_id"": ""S52"",
      ""covered_employee_ids"": [
        4,
        5
      ],
      ""employer_contribution"": 180
    },
    {
      ""bundle_id"": ""S53"",
      ""covered_employee_ids"": [
        3,
        4,
        5,
        7
      ],
      ""employer_contribution"": 168
    },
    {
      ""bundle_id"": ""S54"",
      ""covered_employee_ids"": [
        5,
        7
      ],
      ""employer_contribution"": 88
    },
    {
      ""bundle_id"": ""S55"",
      ""covered_employee_ids"": [
        2,
        3,
        4,
        5
      ],
      ""employer_contribution"": 200
    },
    {
      ""bundle_id"": ""S56"",
      ""covered_employee_ids"": [
        12,
        13,
        14,
        15
      ],
      ""employer_contribution"": 136
    },
    {
      ""bundle_id"": ""S57"",
      ""covered_employee_ids"": [
        11,
        12,
        14,
        15,
        16
      ],
      ""employer_contribution"": 225
    },
    {
      ""bundle_id"": ""S58"",
      ""covered_employee_ids"": [
        2,
        3,
        5,
        6
      ],
      ""employer_contribution"": 68
    },
    {
      ""bundle_id"": ""S59"",
      ""covered_employee_ids"": [
        9,
        13,
        14,
        15
      ],
      ""employer_contribution"": 220
    },
    {
      ""bundle_id"": ""S60"",
      ""covered_employee_ids"": [
        1
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S61"",
      ""covered_employee_ids"": [
        2
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S62"",
      ""covered_employee_ids"": [
        3
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S63"",
      ""covered_employee_ids"": [
        4
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S64"",
      ""covered_employee_ids"": [
        5
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S65"",
      ""covered_employee_ids"": [
        6
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S66"",
      ""covered_employee_ids"": [
        7
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S67"",
      ""covered_employee_ids"": [
        8
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S68"",
      ""covered_employee_ids"": [
        9
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S69"",
      ""covered_employee_ids"": [
        10
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S70"",
      ""covered_employee_ids"": [
        11
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S71"",
      ""covered_employee_ids"": [
        12
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S72"",
      ""covered_employee_ids"": [
        13
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S73"",
      ""covered_employee_ids"": [
        14
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S74"",
      ""covered_employee_ids"": [
        15
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S75"",
      ""covered_employee_ids"": [
        16
      ],
      ""employer_contribution"": 10000
    },
    {
      ""bundle_id"": ""S76"",
      ""covered_employee_ids"": [
        17
      ],
      ""employer_contribution"": 10000
    }
  ]
}

You can reply using this simple JSON shape so it's easy to parse — just put the package IDs you pick into the array under ""solution"".

{
  ""solution"": [""package_id"", ...]
}

This is just a sketch: ""solution"" should hold the IDs of the benefit packages you want to use (one ID per selected package). Think of it like a tiny form: drop the exact package identifiers from the instance into that array and you're done.

Please use the identifiers exactly as they appear in the instance input — no renaming and no new labels.
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 17, 'num_sets': 76, 'density': 0.17105263157894737, 'sets': [{'id': 1, 'elements': [7, 8, 9, 10, 11], 'cost': 420}, {'id': 2, 'elements': [14, 16, 17], 'cost': 282}, {'id': 3, 'elements': [2, 4], 'cost': 34}, {'id': 4, 'elements': [4, 5, 6, 9], 'cost': 60}, {'id': 5, 'elements': [14, 15], 'cost': 136}, {'id': 6, 'elements': [13, 14, 15], 'cost': 114}, {'id': 7, 'elements': [8, 9, 10, 11], 'cost': 76}, {'id': 8, 'elements': [6, 8, 10], 'cost': 234}, {'id': 9, 'elements': [14, 15, 16], 'cost': 54}, {'id': 10, 'elements': [12, 15], 'cost': 40}, {'id': 11, 'elements': [1, 2, 4], 'cost': 207}, {'id': 12, 'elements': [2, 3, 4], 'cost': 135}, {'id': 13, 'elements': [4, 6, 7, 8], 'cost': 220}, {'id': 14, 'elements': [9, 10, 11, 13], 'cost': 332}, {'id': 15, 'elements': [1, 2, 3, 4], 'cost': 292}, {'id': 16, 'elements': [16, 17], 'cost': 86}, {'id': 17, 'elements': [9, 10, 11, 12], 'cost': 108}, {'id': 18, 'elements': [6, 12], 'cost': 158}, {'id': 19, 'elements': [1, 6], 'cost': 20}, {'id': 20, 'elements': [1, 2, 3, 4, 5], 'cost': 490}, {'id': 21, 'elements': [12, 13], 'cost': 122}, {'id': 22, 'elements': [7, 9], 'cost': 184}, {'id': 23, 'elements': [13, 14], 'cost': 66}, {'id': 24, 'elements': [12, 14, 15, 16], 'cost': 396}, {'id': 25, 'elements': [13, 15, 16, 17], 'cost': 196}, {'id': 26, 'elements': [1, 2, 3], 'cost': 228}, {'id': 27, 'elements': [3, 5, 6], 'cost': 6}, {'id': 28, 'elements': [1, 2, 3, 7], 'cost': 364}, {'id': 29, 'elements': [3, 4, 5], 'cost': 183}, {'id': 30, 'elements': [13, 14, 15, 16, 17], 'cost': 25}, {'id': 31, 'elements': [3, 6, 7], 'cost': 33}, {'id': 32, 'elements': [1, 2, 5], 'cost': 39}, {'id': 33, 'elements': [5, 8, 10, 11, 13], 'cost': 205}, {'id': 34, 'elements': [5, 6, 7, 8, 9], 'cost': 260}, {'id': 35, 'elements': [1, 3, 4], 'cost': 282}, {'id': 36, 'elements': [4, 5, 6], 'cost': 195}, {'id': 37, 'elements': [3, 4, 5, 7, 8], 'cost': 160}, {'id': 38, 'elements': [5, 7, 8], 'cost': 147}, {'id': 39, 'elements': [5, 6, 7, 9], 'cost': 140}, {'id': 40, 'elements': [3, 5, 6, 7, 9], 'cost': 235}, {'id': 41, 'elements': [15, 16, 17], 'cost': 15}, {'id': 42, 'elements': [11, 13, 14, 15, 17], 'cost': 440}, {'id': 43, 'elements': [6, 7, 9, 10], 'cost': 140}, {'id': 44, 'elements': [6, 7], 'cost': 10}, {'id': 45, 'elements': [12, 13, 15, 16], 'cost': 308}, {'id': 46, 'elements': [12, 15, 16], 'cost': 294}, {'id': 47, 'elements': [7, 8], 'cost': 180}, {'id': 48, 'elements': [5, 7, 8, 9, 10], 'cost': 35}, {'id': 49, 'elements': [8, 9, 10, 11, 12], 'cost': 425}, {'id': 50, 'elements': [3, 5], 'cost': 62}, {'id': 51, 'elements': [12, 13, 14, 15, 16], 'cost': 310}, {'id': 52, 'elements': [4, 5], 'cost': 180}, {'id': 53, 'elements': [3, 4, 5, 7], 'cost': 168}, {'id': 54, 'elements': [5, 7], 'cost': 88}, {'id': 55, 'elements': [2, 3, 4, 5], 'cost': 200}, {'id': 56, 'elements': [12, 13, 14, 15], 'cost': 136}, {'id': 57, 'elements': [11, 12, 14, 15, 16], 'cost': 225}, {'id': 58, 'elements': [2, 3, 5, 6], 'cost': 68}, {'id': 59, 'elements': [9, 13, 14, 15], 'cost': 220}, {'id': 60, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 66, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 67, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 68, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 69, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 70, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 71, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 72, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 73, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 74, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 75, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 76, 'elements': [17], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0033_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0033_bag_stats.png'}","[3, 17, 19, 30, 47, 50]",429.0,"{'num_elements': 17, 'num_sets': 76, 'sets': [{'id': 'S1', 'elements': [7, 8, 9, 10, 11], 'cost': 420}, {'id': 'S2', 'elements': [14, 16, 17], 'cost': 282}, {'id': 'S3', 'elements': [2, 4], 'cost': 34}, {'id': 'S4', 'elements': [4, 5, 6, 9], 'cost': 60}, {'id': 'S5', 'elements': [14, 15], 'cost': 136}, {'id': 'S6', 'elements': [13, 14, 15], 'cost': 114}, {'id': 'S7', 'elements': [8, 9, 10, 11], 'cost': 76}, {'id': 'S8', 'elements': [6, 8, 10], 'cost': 234}, {'id': 'S9', 'elements': [14, 15, 16], 'cost': 54}, {'id': 'S10', 'elements': [12, 15], 'cost': 40}, {'id': 'S11', 'elements': [1, 2, 4], 'cost': 207}, {'id': 'S12', 'elements': [2, 3, 4], 'cost': 135}, {'id': 'S13', 'elements': [4, 6, 7, 8], 'cost': 220}, {'id': 'S14', 'elements': [9, 10, 11, 13], 'cost': 332}, {'id': 'S15', 'elements': [1, 2, 3, 4], 'cost': 292}, {'id': 'S16', 'elements': [16, 17], 'cost': 86}, {'id': 'S17', 'elements': [9, 10, 11, 12], 'cost': 108}, {'id': 'S18', 'elements': [6, 12], 'cost': 158}, {'id': 'S19', 'elements': [1, 6], 'cost': 20}, {'id': 'S20', 'elements': [1, 2, 3, 4, 5], 'cost': 490}, {'id': 'S21', 'elements': [12, 13], 'cost': 122}, {'id': 'S22', 'elements': [7, 9], 'cost': 184}, {'id': 'S23', 'elements': [13, 14], 'cost': 66}, {'id': 'S24', 'elements': [12, 14, 15, 16], 'cost': 396}, {'id': 'S25', 'elements': [13, 15, 16, 17], 'cost': 196}, {'id': 'S26', 'elements': [1, 2, 3], 'cost': 228}, {'id': 'S27', 'elements': [3, 5, 6], 'cost': 6}, {'id': 'S28', 'elements': [1, 2, 3, 7], 'cost': 364}, {'id': 'S29', 'elements': [3, 4, 5], 'cost': 183}, {'id': 'S30', 'elements': [13, 14, 15, 16, 17], 'cost': 25}, {'id': 'S31', 'elements': [3, 6, 7], 'cost': 33}, {'id': 'S32', 'elements': [1, 2, 5], 'cost': 39}, {'id': 'S33', 'elements': [5, 8, 10, 11, 13], 'cost': 205}, {'id': 'S34', 'elements': [5, 6, 7, 8, 9], 'cost': 260}, {'id': 'S35', 'elements': [1, 3, 4], 'cost': 282}, {'id': 'S36', 'elements': [4, 5, 6], 'cost': 195}, {'id': 'S37', 'elements': [3, 4, 5, 7, 8], 'cost': 160}, {'id': 'S38', 'elements': [5, 7, 8], 'cost': 147}, {'id': 'S39', 'elements': [5, 6, 7, 9], 'cost': 140}, {'id': 'S40', 'elements': [3, 5, 6, 7, 9], 'cost': 235}, {'id': 'S41', 'elements': [15, 16, 17], 'cost': 15}, {'id': 'S42', 'elements': [11, 13, 14, 15, 17], 'cost': 440}, {'id': 'S43', 'elements': [6, 7, 9, 10], 'cost': 140}, {'id': 'S44', 'elements': [6, 7], 'cost': 10}, {'id': 'S45', 'elements': [12, 13, 15, 16], 'cost': 308}, {'id': 'S46', 'elements': [12, 15, 16], 'cost': 294}, {'id': 'S47', 'elements': [7, 8], 'cost': 180}, {'id': 'S48', 'elements': [5, 7, 8, 9, 10], 'cost': 35}, {'id': 'S49', 'elements': [8, 9, 10, 11, 12], 'cost': 425}, {'id': 'S50', 'elements': [3, 5], 'cost': 62}, {'id': 'S51', 'elements': [12, 13, 14, 15, 16], 'cost': 310}, {'id': 'S52', 'elements': [4, 5], 'cost': 180}, {'id': 'S53', 'elements': [3, 4, 5, 7], 'cost': 168}, {'id': 'S54', 'elements': [5, 7], 'cost': 88}, {'id': 'S55', 'elements': [2, 3, 4, 5], 'cost': 200}, {'id': 'S56', 'elements': [12, 13, 14, 15], 'cost': 136}, {'id': 'S57', 'elements': [11, 12, 14, 15, 16], 'cost': 225}, {'id': 'S58', 'elements': [2, 3, 5, 6], 'cost': 68}, {'id': 'S59', 'elements': [9, 13, 14, 15], 'cost': 220}, {'id': 'S60', 'elements': [1], 'cost': 10000}, {'id': 'S61', 'elements': [2], 'cost': 10000}, {'id': 'S62', 'elements': [3], 'cost': 10000}, {'id': 'S63', 'elements': [4], 'cost': 10000}, {'id': 'S64', 'elements': [5], 'cost': 10000}, {'id': 'S65', 'elements': [6], 'cost': 10000}, {'id': 'S66', 'elements': [7], 'cost': 10000}, {'id': 'S67', 'elements': [8], 'cost': 10000}, {'id': 'S68', 'elements': [9], 'cost': 10000}, {'id': 'S69', 'elements': [10], 'cost': 10000}, {'id': 'S70', 'elements': [11], 'cost': 10000}, {'id': 'S71', 'elements': [12], 'cost': 10000}, {'id': 'S72', 'elements': [13], 'cost': 10000}, {'id': 'S73', 'elements': [14], 'cost': 10000}, {'id': 'S74', 'elements': [15], 'cost': 10000}, {'id': 'S75', 'elements': [16], 'cost': 10000}, {'id': 'S76', 'elements': [17], 'cost': 10000}]}","['S3', 'S17', 'S19', 'S30', 'S47', 'S50']",34,json,1
SPP,SPP,"At my job we’re choosing from several route contracts, and the rule is simple: every delivery stop must belong to exactly one chosen contract — nothing left out and nothing covered twice. Each contract charges a fee, so better choices are those with the lowest combined bill; you get that combined bill by adding together the fees of all picked contracts. The concrete instance details appear below.

- **num_stops**: 13
- **num_contracts_available**: 52

| contract_id | contract_fee | stops_covered |
|---|---|---|
| S1 | 116 | 8 11 |
| S2 | 141 | 10 11 12 |
| S3 | 100 | 2 5 |
| S4 | 104 | 0 1 2 4 |
| S5 | 111 | 6 9 10 |
| S6 | 7 | 7 |
| S7 | 16 | 9 10 11 12 |
| S8 | 117 | 1 2 3 |
| S9 | 153 | 7 9 10 |
| S10 | 38 | 0 1 |
| S11 | 240 | 5 6 8 |
| S12 | 12 | 11 12 |
| S13 | 78 | 3 |
| S14 | 176 | 8 9 10 11 |
| S15 | 158 | 3 6 |
| S16 | 48 | 8 9 |
| S17 | 255 | 1 3 5 |
| S18 | 164 | 4 5 |
| S19 | 58 | 2 4 |
| S20 | 78 | 5 |
| S21 | 183 | 8 9 10 |
| S22 | 384 | 0 1 2 3 |
| S23 | 15 | 7 8 9 |
| S24 | 135 | 8 9 11 |
| S25 | 76 | 10 12 |
| S26 | 66 | 4 7 8 |
| S27 | 111 | 6 9 12 |
| S28 | 147 | 7 8 10 |
| S29 | 222 | 0 1 5 |
| S30 | 104 | 3 5 |
| S31 | 123 | 4 5 6 |
| S32 | 294 | 8 11 12 |
| S33 | 69 | 5 6 9 |
| S34 | 162 | 0 2 3 |
| S35 | 260 | 3 5 6 7 |
| S36 | 88 | 6 8 9 10 |
| S37 | 231 | 4 8 9 |
| S38 | 236 | 8 9 10 12 |
| S39 | 236 | 2 3 4 6 |
| S40 | 10000 | 0 |
| S41 | 10000 | 1 |
| S42 | 10000 | 2 |
| S43 | 10000 | 3 |
| S44 | 10000 | 4 |
| S45 | 10000 | 5 |
| S46 | 10000 | 6 |
| S47 | 10000 | 7 |
| S48 | 10000 | 8 |
| S49 | 10000 | 9 |
| S50 | 10000 | 10 |
| S51 | 10000 | 11 |
| S52 | 10000 | 12 |

Pretty simple — when you send back the selected contracts, just stick to this little JSON shape so it's easy to parse:

{
  ""solution"": [""contract_id"", ...]
}

Here ""solution"" is the list of the route contracts you pick (one entry per chosen contract). The placeholder ""contract_id"" is just a stand-in — in your actual reply put the real contract identifiers from the instance. This block is just a sketch of the expected shape, not the final answer.

Please be sure to use the identifiers exactly as they appear in the instance input — don't rename them or invent new labels. Valid identifiers look like plain numbers such as ""1"" or ""23"", single capital letters like ""A"" or ""B"", or a capital letter followed by digits like ""A1"" or ""X7"".","{'problem_type': 'SPP', 'num_elements': 13, 'num_sets': 52, 'density': 0.1804733727810651, 'sets': [{'id': 1, 'elements': [9, 12], 'cost': 116}, {'id': 2, 'elements': [11, 12, 13], 'cost': 141}, {'id': 3, 'elements': [3, 6], 'cost': 100}, {'id': 4, 'elements': [1, 2, 3, 5], 'cost': 104}, {'id': 5, 'elements': [7, 10, 11], 'cost': 111}, {'id': 6, 'elements': [8], 'cost': 7}, {'id': 7, 'elements': [10, 11, 12, 13], 'cost': 16}, {'id': 8, 'elements': [2, 3, 4], 'cost': 117}, {'id': 9, 'elements': [8, 10, 11], 'cost': 153}, {'id': 10, 'elements': [1, 2], 'cost': 38}, {'id': 11, 'elements': [6, 7, 9], 'cost': 240}, {'id': 12, 'elements': [12, 13], 'cost': 12}, {'id': 13, 'elements': [4], 'cost': 78}, {'id': 14, 'elements': [9, 10, 11, 12], 'cost': 176}, {'id': 15, 'elements': [4, 7], 'cost': 158}, {'id': 16, 'elements': [9, 10], 'cost': 48}, {'id': 17, 'elements': [2, 4, 6], 'cost': 255}, {'id': 18, 'elements': [5, 6], 'cost': 164}, {'id': 19, 'elements': [3, 5], 'cost': 58}, {'id': 20, 'elements': [6], 'cost': 78}, {'id': 21, 'elements': [9, 10, 11], 'cost': 183}, {'id': 22, 'elements': [1, 2, 3, 4], 'cost': 384}, {'id': 23, 'elements': [8, 9, 10], 'cost': 15}, {'id': 24, 'elements': [9, 10, 12], 'cost': 135}, {'id': 25, 'elements': [11, 13], 'cost': 76}, {'id': 26, 'elements': [5, 8, 9], 'cost': 66}, {'id': 27, 'elements': [7, 10, 13], 'cost': 111}, {'id': 28, 'elements': [8, 9, 11], 'cost': 147}, {'id': 29, 'elements': [1, 2, 6], 'cost': 222}, {'id': 30, 'elements': [4, 6], 'cost': 104}, {'id': 31, 'elements': [5, 6, 7], 'cost': 123}, {'id': 32, 'elements': [9, 12, 13], 'cost': 294}, {'id': 33, 'elements': [6, 7, 10], 'cost': 69}, {'id': 34, 'elements': [1, 3, 4], 'cost': 162}, {'id': 35, 'elements': [4, 6, 7, 8], 'cost': 260}, {'id': 36, 'elements': [7, 9, 10, 11], 'cost': 88}, {'id': 37, 'elements': [5, 9, 10], 'cost': 231}, {'id': 38, 'elements': [9, 10, 11, 13], 'cost': 236}, {'id': 39, 'elements': [3, 4, 5, 7], 'cost': 236}, {'id': 40, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 41, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 42, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 43, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 44, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 45, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 46, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 47, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [13], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0034_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0034_bag_stats.png'}","[6, 10, 12, 19, 30, 36]",307.0,"{'num_elements': 13, 'num_sets': 52, 'sets': [{'id': 'S1', 'elements': [8, 11], 'cost': 116}, {'id': 'S2', 'elements': [10, 11, 12], 'cost': 141}, {'id': 'S3', 'elements': [2, 5], 'cost': 100}, {'id': 'S4', 'elements': [0, 1, 2, 4], 'cost': 104}, {'id': 'S5', 'elements': [6, 9, 10], 'cost': 111}, {'id': 'S6', 'elements': [7], 'cost': 7}, {'id': 'S7', 'elements': [9, 10, 11, 12], 'cost': 16}, {'id': 'S8', 'elements': [1, 2, 3], 'cost': 117}, {'id': 'S9', 'elements': [7, 9, 10], 'cost': 153}, {'id': 'S10', 'elements': [0, 1], 'cost': 38}, {'id': 'S11', 'elements': [5, 6, 8], 'cost': 240}, {'id': 'S12', 'elements': [11, 12], 'cost': 12}, {'id': 'S13', 'elements': [3], 'cost': 78}, {'id': 'S14', 'elements': [8, 9, 10, 11], 'cost': 176}, {'id': 'S15', 'elements': [3, 6], 'cost': 158}, {'id': 'S16', 'elements': [8, 9], 'cost': 48}, {'id': 'S17', 'elements': [1, 3, 5], 'cost': 255}, {'id': 'S18', 'elements': [4, 5], 'cost': 164}, {'id': 'S19', 'elements': [2, 4], 'cost': 58}, {'id': 'S20', 'elements': [5], 'cost': 78}, {'id': 'S21', 'elements': [8, 9, 10], 'cost': 183}, {'id': 'S22', 'elements': [0, 1, 2, 3], 'cost': 384}, {'id': 'S23', 'elements': [7, 8, 9], 'cost': 15}, {'id': 'S24', 'elements': [8, 9, 11], 'cost': 135}, {'id': 'S25', 'elements': [10, 12], 'cost': 76}, {'id': 'S26', 'elements': [4, 7, 8], 'cost': 66}, {'id': 'S27', 'elements': [6, 9, 12], 'cost': 111}, {'id': 'S28', 'elements': [7, 8, 10], 'cost': 147}, {'id': 'S29', 'elements': [0, 1, 5], 'cost': 222}, {'id': 'S30', 'elements': [3, 5], 'cost': 104}, {'id': 'S31', 'elements': [4, 5, 6], 'cost': 123}, {'id': 'S32', 'elements': [8, 11, 12], 'cost': 294}, {'id': 'S33', 'elements': [5, 6, 9], 'cost': 69}, {'id': 'S34', 'elements': [0, 2, 3], 'cost': 162}, {'id': 'S35', 'elements': [3, 5, 6, 7], 'cost': 260}, {'id': 'S36', 'elements': [6, 8, 9, 10], 'cost': 88}, {'id': 'S37', 'elements': [4, 8, 9], 'cost': 231}, {'id': 'S38', 'elements': [8, 9, 10, 12], 'cost': 236}, {'id': 'S39', 'elements': [2, 3, 4, 6], 'cost': 236}, {'id': 'S40', 'elements': [0], 'cost': 10000}, {'id': 'S41', 'elements': [1], 'cost': 10000}, {'id': 'S42', 'elements': [2], 'cost': 10000}, {'id': 'S43', 'elements': [3], 'cost': 10000}, {'id': 'S44', 'elements': [4], 'cost': 10000}, {'id': 'S45', 'elements': [5], 'cost': 10000}, {'id': 'S46', 'elements': [6], 'cost': 10000}, {'id': 'S47', 'elements': [7], 'cost': 10000}, {'id': 'S48', 'elements': [8], 'cost': 10000}, {'id': 'S49', 'elements': [9], 'cost': 10000}, {'id': 'S50', 'elements': [10], 'cost': 10000}, {'id': 'S51', 'elements': [11], 'cost': 10000}, {'id': 'S52', 'elements': [12], 'cost': 10000}]}","['S6', 'S10', 'S12', 'S19', 'S30', 'S36']",35,markdown_table,0
SPP,SPP,"We had a checklist of customer wants and a pile of possible feature modules, and the task was to choose some modules so every want is handled by a single module — no gaps and no overlaps. The better choice is the cheaper one overall: total development cost is simply the sum of the costs of the modules that get picked, and the goal is to make that sum as small as possible. The exact list of requirements and module costs follows below.

{
  ""num_requirements"": 14,
  ""num_modules"": 61,
  ""sets"": [
    {
      ""module_id"": ""S1"",
      ""covered_requirements"": [
        1,
        2,
        3
      ],
      ""development_cost"": 63
    },
    {
      ""module_id"": ""S2"",
      ""covered_requirements"": [
        8,
        13
      ],
      ""development_cost"": 10
    },
    {
      ""module_id"": ""S3"",
      ""covered_requirements"": [
        7,
        8,
        10
      ],
      ""development_cost"": 30
    },
    {
      ""module_id"": ""S4"",
      ""covered_requirements"": [
        3,
        4,
        5,
        6
      ],
      ""development_cost"": 276
    },
    {
      ""module_id"": ""S5"",
      ""covered_requirements"": [
        6,
        7,
        8
      ],
      ""development_cost"": 153
    },
    {
      ""module_id"": ""S6"",
      ""covered_requirements"": [
        9,
        10,
        11,
        12
      ],
      ""development_cost"": 336
    },
    {
      ""module_id"": ""S7"",
      ""covered_requirements"": [
        10,
        11,
        12
      ],
      ""development_cost"": 225
    },
    {
      ""module_id"": ""S8"",
      ""covered_requirements"": [
        0,
        1,
        2,
        3
      ],
      ""development_cost"": 128
    },
    {
      ""module_id"": ""S9"",
      ""covered_requirements"": [
        1,
        2,
        3,
        4
      ],
      ""development_cost"": 364
    },
    {
      ""module_id"": ""S10"",
      ""covered_requirements"": [
        2,
        3,
        4
      ],
      ""development_cost"": 129
    },
    {
      ""module_id"": ""S11"",
      ""covered_requirements"": [
        8,
        9,
        10,
        11
      ],
      ""development_cost"": 268
    },
    {
      ""module_id"": ""S12"",
      ""covered_requirements"": [
        3,
        5
      ],
      ""development_cost"": 184
    },
    {
      ""module_id"": ""S13"",
      ""covered_requirements"": [
        10,
        12
      ],
      ""development_cost"": 122
    },
    {
      ""module_id"": ""S14"",
      ""covered_requirements"": [
        9,
        10,
        11
      ],
      ""development_cost"": 57
    },
    {
      ""module_id"": ""S15"",
      ""covered_requirements"": [
        9,
        12,
        13
      ],
      ""development_cost"": 84
    },
    {
      ""module_id"": ""S16"",
      ""covered_requirements"": [
        1,
        2,
        4
      ],
      ""development_cost"": 288
    },
    {
      ""module_id"": ""S17"",
      ""covered_requirements"": [
        6,
        7,
        8,
        9
      ],
      ""development_cost"": 80
    },
    {
      ""module_id"": ""S18"",
      ""covered_requirements"": [
        9,
        12
      ],
      ""development_cost"": 80
    },
    {
      ""module_id"": ""S19"",
      ""covered_requirements"": [
        1,
        3
      ],
      ""development_cost"": 32
    },
    {
      ""module_id"": ""S20"",
      ""covered_requirements"": [
        4,
        5,
        6
      ],
      ""development_cost"": 6
    },
    {
      ""module_id"": ""S21"",
      ""covered_requirements"": [
        10,
        11,
        12,
        13
      ],
      ""development_cost"": 188
    },
    {
      ""module_id"": ""S22"",
      ""covered_requirements"": [
        5,
        6,
        8
      ],
      ""development_cost"": 105
    },
    {
      ""module_id"": ""S23"",
      ""covered_requirements"": [
        6,
        8,
        9
      ],
      ""development_cost"": 24
    },
    {
      ""module_id"": ""S24"",
      ""covered_requirements"": [
        8,
        9
      ],
      ""development_cost"": 192
    },
    {
      ""module_id"": ""S25"",
      ""covered_requirements"": [
        12,
        13
      ],
      ""development_cost"": 34
    },
    {
      ""module_id"": ""S26"",
      ""covered_requirements"": [
        0,
        1,
        3
      ],
      ""development_cost"": 186
    },
    {
      ""module_id"": ""S27"",
      ""covered_requirements"": [
        6,
        9,
        10
      ],
      ""development_cost"": 30
    },
    {
      ""module_id"": ""S28"",
      ""covered_requirements"": [
        1,
        2
      ],
      ""development_cost"": 26
    },
    {
      ""module_id"": ""S29"",
      ""covered_requirements"": [
        0,
        2,
        3
      ],
      ""development_cost"": 168
    },
    {
      ""module_id"": ""S30"",
      ""covered_requirements"": [
        7,
        12,
        13
      ],
      ""development_cost"": 231
    },
    {
      ""module_id"": ""S31"",
      ""covered_requirements"": [
        9,
        10
      ],
      ""development_cost"": 132
    },
    {
      ""module_id"": ""S32"",
      ""covered_requirements"": [
        4,
        5
      ],
      ""development_cost"": 162
    },
    {
      ""module_id"": ""S33"",
      ""covered_requirements"": [
        10,
        11,
        13
      ],
      ""development_cost"": 198
    },
    {
      ""module_id"": ""S34"",
      ""covered_requirements"": [
        0,
        1,
        2
      ],
      ""development_cost"": 120
    },
    {
      ""module_id"": ""S35"",
      ""covered_requirements"": [
        1,
        2,
        3,
        5
      ],
      ""development_cost"": 88
    },
    {
      ""module_id"": ""S36"",
      ""covered_requirements"": [
        7,
        8,
        9,
        10
      ],
      ""development_cost"": 308
    },
    {
      ""module_id"": ""S37"",
      ""covered_requirements"": [
        4,
        5,
        6,
        7
      ],
      ""development_cost"": 232
    },
    {
      ""module_id"": ""S38"",
      ""covered_requirements"": [
        1,
        3,
        5
      ],
      ""development_cost"": 276
    },
    {
      ""module_id"": ""S39"",
      ""covered_requirements"": [
        11,
        12
      ],
      ""development_cost"": 98
    },
    {
      ""module_id"": ""S40"",
      ""covered_requirements"": [
        7,
        9
      ],
      ""development_cost"": 12
    },
    {
      ""module_id"": ""S41"",
      ""covered_requirements"": [
        4,
        5,
        7,
        8
      ],
      ""development_cost"": 312
    },
    {
      ""module_id"": ""S42"",
      ""covered_requirements"": [
        2,
        3
      ],
      ""development_cost"": 192
    },
    {
      ""module_id"": ""S43"",
      ""covered_requirements"": [
        11,
        12,
        13
      ],
      ""development_cost"": 267
    },
    {
      ""module_id"": ""S44"",
      ""covered_requirements"": [
        13
      ],
      ""development_cost"": 90
    },
    {
      ""module_id"": ""S45"",
      ""covered_requirements"": [
        4,
        6,
        7
      ],
      ""development_cost"": 288
    },
    {
      ""module_id"": ""S46"",
      ""covered_requirements"": [
        6,
        7,
        8,
        10
      ],
      ""development_cost"": 172
    },
    {
      ""module_id"": ""S47"",
      ""covered_requirements"": [
        9,
        10,
        12
      ],
      ""development_cost"": 45
    },
    {
      ""module_id"": ""S48"",
      ""covered_requirements"": [
        0
      ],
      ""development_cost"": 10000
    },
    {
      ""module_id"": ""S49"",
      ""covered_requirements"": [
        1
      ],
      ""development_cost"": 10000
    },
    {
      ""module_id"": ""S50"",
      ""covered_requirements"": [
        2
      ],
      ""development_cost"": 10000
    },
    {
      ""module_id"": ""S51"",
      ""covered_requirements"": [
        3
      ],
      ""development_cost"": 10000
    },
    {
      ""module_id"": ""S52"",
      ""covered_requirements"": [
        4
      ],
      ""development_cost"": 10000
    },
    {
      ""module_id"": ""S53"",
      ""covered_requirements"": [
        5
      ],
      ""development_cost"": 10000
    },
    {
      ""module_id"": ""S54"",
      ""covered_requirements"": [
        6
      ],
      ""development_cost"": 10000
    },
    {
      ""module_id"": ""S55"",
      ""covered_requirements"": [
        7
      ],
      ""development_cost"": 10000
    },
    {
      ""module_id"": ""S56"",
      ""covered_requirements"": [
        8
      ],
      ""development_cost"": 10000
    },
    {
      ""module_id"": ""S57"",
      ""covered_requirements"": [
        9
      ],
      ""development_cost"": 10000
    },
    {
      ""module_id"": ""S58"",
      ""covered_requirements"": [
        10
      ],
      ""development_cost"": 10000
    },
    {
      ""module_id"": ""S59"",
      ""covered_requirements"": [
        11
      ],
      ""development_cost"": 10000
    },
    {
      ""module_id"": ""S60"",
      ""covered_requirements"": [
        12
      ],
      ""development_cost"": 10000
    },
    {
      ""module_id"": ""S61"",
      ""covered_requirements"": [
        13
      ],
      ""development_cost"": 10000
    }
  ]
}

Also, just to keep things tidy when you send the answer back, please use this simple JSON layout so I can read it automatically.

{
  ""solution"": [""module_id"", ...]
}

""solution"" is the list of chosen modules (one entry per module). The array items are placeholders showing where the actual module identifiers go — replace ""module_id"" with the real IDs from the instance when you submit the final selection. This is just a sketch of the shape I expect, not the final answer.

One more thing: all identifiers must be used exactly as they appear in the instance input — do not rename them or invent new labels.
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 14, 'num_sets': 61, 'density': 0.17798594847775176, 'sets': [{'id': 1, 'elements': [2, 3, 4], 'cost': 63}, {'id': 2, 'elements': [9, 14], 'cost': 10}, {'id': 3, 'elements': [8, 9, 11], 'cost': 30}, {'id': 4, 'elements': [4, 5, 6, 7], 'cost': 276}, {'id': 5, 'elements': [7, 8, 9], 'cost': 153}, {'id': 6, 'elements': [10, 11, 12, 13], 'cost': 336}, {'id': 7, 'elements': [11, 12, 13], 'cost': 225}, {'id': 8, 'elements': [1, 2, 3, 4], 'cost': 128}, {'id': 9, 'elements': [2, 3, 4, 5], 'cost': 364}, {'id': 10, 'elements': [3, 4, 5], 'cost': 129}, {'id': 11, 'elements': [9, 10, 11, 12], 'cost': 268}, {'id': 12, 'elements': [4, 6], 'cost': 184}, {'id': 13, 'elements': [11, 13], 'cost': 122}, {'id': 14, 'elements': [10, 11, 12], 'cost': 57}, {'id': 15, 'elements': [10, 13, 14], 'cost': 84}, {'id': 16, 'elements': [2, 3, 5], 'cost': 288}, {'id': 17, 'elements': [7, 8, 9, 10], 'cost': 80}, {'id': 18, 'elements': [10, 13], 'cost': 80}, {'id': 19, 'elements': [2, 4], 'cost': 32}, {'id': 20, 'elements': [5, 6, 7], 'cost': 6}, {'id': 21, 'elements': [11, 12, 13, 14], 'cost': 188}, {'id': 22, 'elements': [6, 7, 9], 'cost': 105}, {'id': 23, 'elements': [7, 9, 10], 'cost': 24}, {'id': 24, 'elements': [9, 10], 'cost': 192}, {'id': 25, 'elements': [13, 14], 'cost': 34}, {'id': 26, 'elements': [1, 2, 4], 'cost': 186}, {'id': 27, 'elements': [7, 10, 11], 'cost': 30}, {'id': 28, 'elements': [2, 3], 'cost': 26}, {'id': 29, 'elements': [1, 3, 4], 'cost': 168}, {'id': 30, 'elements': [8, 13, 14], 'cost': 231}, {'id': 31, 'elements': [10, 11], 'cost': 132}, {'id': 32, 'elements': [5, 6], 'cost': 162}, {'id': 33, 'elements': [11, 12, 14], 'cost': 198}, {'id': 34, 'elements': [1, 2, 3], 'cost': 120}, {'id': 35, 'elements': [2, 3, 4, 6], 'cost': 88}, {'id': 36, 'elements': [8, 9, 10, 11], 'cost': 308}, {'id': 37, 'elements': [5, 6, 7, 8], 'cost': 232}, {'id': 38, 'elements': [2, 4, 6], 'cost': 276}, {'id': 39, 'elements': [12, 13], 'cost': 98}, {'id': 40, 'elements': [8, 10], 'cost': 12}, {'id': 41, 'elements': [5, 6, 8, 9], 'cost': 312}, {'id': 42, 'elements': [3, 4], 'cost': 192}, {'id': 43, 'elements': [12, 13, 14], 'cost': 267}, {'id': 44, 'elements': [14], 'cost': 90}, {'id': 45, 'elements': [5, 7, 8], 'cost': 288}, {'id': 46, 'elements': [7, 8, 9, 11], 'cost': 172}, {'id': 47, 'elements': [10, 11, 13], 'cost': 45}, {'id': 48, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [14], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0035_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0035_bag_stats.png'}","[2, 7, 8, 20, 40]",381.0,"{'num_elements': 14, 'num_sets': 61, 'sets': [{'id': 'S1', 'elements': [1, 2, 3], 'cost': 63}, {'id': 'S2', 'elements': [8, 13], 'cost': 10}, {'id': 'S3', 'elements': [7, 8, 10], 'cost': 30}, {'id': 'S4', 'elements': [3, 4, 5, 6], 'cost': 276}, {'id': 'S5', 'elements': [6, 7, 8], 'cost': 153}, {'id': 'S6', 'elements': [9, 10, 11, 12], 'cost': 336}, {'id': 'S7', 'elements': [10, 11, 12], 'cost': 225}, {'id': 'S8', 'elements': [0, 1, 2, 3], 'cost': 128}, {'id': 'S9', 'elements': [1, 2, 3, 4], 'cost': 364}, {'id': 'S10', 'elements': [2, 3, 4], 'cost': 129}, {'id': 'S11', 'elements': [8, 9, 10, 11], 'cost': 268}, {'id': 'S12', 'elements': [3, 5], 'cost': 184}, {'id': 'S13', 'elements': [10, 12], 'cost': 122}, {'id': 'S14', 'elements': [9, 10, 11], 'cost': 57}, {'id': 'S15', 'elements': [9, 12, 13], 'cost': 84}, {'id': 'S16', 'elements': [1, 2, 4], 'cost': 288}, {'id': 'S17', 'elements': [6, 7, 8, 9], 'cost': 80}, {'id': 'S18', 'elements': [9, 12], 'cost': 80}, {'id': 'S19', 'elements': [1, 3], 'cost': 32}, {'id': 'S20', 'elements': [4, 5, 6], 'cost': 6}, {'id': 'S21', 'elements': [10, 11, 12, 13], 'cost': 188}, {'id': 'S22', 'elements': [5, 6, 8], 'cost': 105}, {'id': 'S23', 'elements': [6, 8, 9], 'cost': 24}, {'id': 'S24', 'elements': [8, 9], 'cost': 192}, {'id': 'S25', 'elements': [12, 13], 'cost': 34}, {'id': 'S26', 'elements': [0, 1, 3], 'cost': 186}, {'id': 'S27', 'elements': [6, 9, 10], 'cost': 30}, {'id': 'S28', 'elements': [1, 2], 'cost': 26}, {'id': 'S29', 'elements': [0, 2, 3], 'cost': 168}, {'id': 'S30', 'elements': [7, 12, 13], 'cost': 231}, {'id': 'S31', 'elements': [9, 10], 'cost': 132}, {'id': 'S32', 'elements': [4, 5], 'cost': 162}, {'id': 'S33', 'elements': [10, 11, 13], 'cost': 198}, {'id': 'S34', 'elements': [0, 1, 2], 'cost': 120}, {'id': 'S35', 'elements': [1, 2, 3, 5], 'cost': 88}, {'id': 'S36', 'elements': [7, 8, 9, 10], 'cost': 308}, {'id': 'S37', 'elements': [4, 5, 6, 7], 'cost': 232}, {'id': 'S38', 'elements': [1, 3, 5], 'cost': 276}, {'id': 'S39', 'elements': [11, 12], 'cost': 98}, {'id': 'S40', 'elements': [7, 9], 'cost': 12}, {'id': 'S41', 'elements': [4, 5, 7, 8], 'cost': 312}, {'id': 'S42', 'elements': [2, 3], 'cost': 192}, {'id': 'S43', 'elements': [11, 12, 13], 'cost': 267}, {'id': 'S44', 'elements': [13], 'cost': 90}, {'id': 'S45', 'elements': [4, 6, 7], 'cost': 288}, {'id': 'S46', 'elements': [6, 7, 8, 10], 'cost': 172}, {'id': 'S47', 'elements': [9, 10, 12], 'cost': 45}, {'id': 'S48', 'elements': [0], 'cost': 10000}, {'id': 'S49', 'elements': [1], 'cost': 10000}, {'id': 'S50', 'elements': [2], 'cost': 10000}, {'id': 'S51', 'elements': [3], 'cost': 10000}, {'id': 'S52', 'elements': [4], 'cost': 10000}, {'id': 'S53', 'elements': [5], 'cost': 10000}, {'id': 'S54', 'elements': [6], 'cost': 10000}, {'id': 'S55', 'elements': [7], 'cost': 10000}, {'id': 'S56', 'elements': [8], 'cost': 10000}, {'id': 'S57', 'elements': [9], 'cost': 10000}, {'id': 'S58', 'elements': [10], 'cost': 10000}, {'id': 'S59', 'elements': [11], 'cost': 10000}, {'id': 'S60', 'elements': [12], 'cost': 10000}, {'id': 'S61', 'elements': [13], 'cost': 10000}]}","['S2', 'S7', 'S8', 'S20', 'S40']",36,json,0
SPP,SPP,"There’s a renovation on the table where each tile box can cover particular patches of floor; the job is to choose boxes so every patch ends up covered by one box and no two chosen boxes cover the same patch. To compare options, just sum the prices of the boxes you pick and aim for the combination with the smallest total cost. The detailed floor map and box price list follow below.

# num_floor_patches=16
# num_tile_boxes=74
box_id,box_price,covered_patches
S1,475,2 3 4 5 6
S2,88,9 12
S3,240,13 14 16
S4,300,1 2 3 5
S5,184,9 11 12 13
S6,276,2 3 4 5
S7,105,6 7 9
S8,114,4 8
S9,190,10 15
S10,54,2 3
S11,64,2 3 5 6
S12,156,5 7
S13,195,9 10 11
S14,147,5 9 12
S15,36,11 12 13 16
S16,24,12 14 15 16
S17,134,4 7
S18,54,13 16
S19,268,8 9 10 11
S20,260,3 4 5 7 8
S21,244,6 7 8 9
S22,98,2 7
S23,36,3 4 5 8
S24,138,15 16
S25,51,1 2 4
S26,192,10 13 15
S27,340,5 6 7 9
S28,158,3 5
S29,78,7 9
S30,105,14 15 16
S31,20,3 4 5 7
S32,328,9 13 15 16
S33,148,6 8
S34,156,11 14 15
S35,288,6 7 8
S36,66,10 13
S37,132,13 15 16
S38,68,7 9 10 12
S39,44,4 5 7 9
S40,145,4 6 8 9 11
S41,272,4 6 7 8
S42,66,12 13 16
S43,87,4 5 6
S44,108,12 15
S45,364,12 13 14 16
S46,42,12 14
S47,174,11 14
S48,84,13 14
S49,198,4 5
S50,316,13 14 15 16
S51,105,3 4 5
S52,455,1 2 3 4 5
S53,224,11 12 13 14
S54,108,3 5 6
S55,52,8 11 13 15
S56,249,10 11 15
S57,270,2 3 5 6 7
S58,268,1 2 3 4
S59,10000,1
S60,10000,2
S61,10000,3
S62,10000,4
S63,10000,5
S64,10000,6
S65,10000,7
S66,10000,8
S67,10000,9
S68,10000,10
S69,10000,11
S70,10000,12
S71,10000,13
S72,10000,14
S73,10000,15
S74,10000,16

Whenever you’re ready to give the chosen boxes, just drop them into a tiny JSON snippet like this so it’s easy to read and machine-friendly:

{
  ""solution"": [""box_id"", ...]
}

Here ""solution"" is the list of the box identifiers you picked to cover every patch exactly once. The ""box_id"" entries are placeholders — replace each one with the exact box label from the instance input when you submit your final choice. This JSON is just a sketch of the shape I’m expecting, not the final answer itself.

Please make sure you use the identifiers exactly as they appear in the instance input — no renaming, no new labels. Valid identifiers look like plain numbers such as ""1"" or ""23"", single capital letters like ""A"" or ""B"", or a capital letter followed by digits like ""A1"" or ""X7"".","{'problem_type': 'SPP', 'num_elements': 16, 'num_sets': 74, 'density': 0.17060810810810811, 'sets': [{'id': 1, 'elements': [2, 3, 4, 5, 6], 'cost': 475}, {'id': 2, 'elements': [9, 12], 'cost': 88}, {'id': 3, 'elements': [13, 14, 16], 'cost': 240}, {'id': 4, 'elements': [1, 2, 3, 5], 'cost': 300}, {'id': 5, 'elements': [9, 11, 12, 13], 'cost': 184}, {'id': 6, 'elements': [2, 3, 4, 5], 'cost': 276}, {'id': 7, 'elements': [6, 7, 9], 'cost': 105}, {'id': 8, 'elements': [4, 8], 'cost': 114}, {'id': 9, 'elements': [10, 15], 'cost': 190}, {'id': 10, 'elements': [2, 3], 'cost': 54}, {'id': 11, 'elements': [2, 3, 5, 6], 'cost': 64}, {'id': 12, 'elements': [5, 7], 'cost': 156}, {'id': 13, 'elements': [9, 10, 11], 'cost': 195}, {'id': 14, 'elements': [5, 9, 12], 'cost': 147}, {'id': 15, 'elements': [11, 12, 13, 16], 'cost': 36}, {'id': 16, 'elements': [12, 14, 15, 16], 'cost': 24}, {'id': 17, 'elements': [4, 7], 'cost': 134}, {'id': 18, 'elements': [13, 16], 'cost': 54}, {'id': 19, 'elements': [8, 9, 10, 11], 'cost': 268}, {'id': 20, 'elements': [3, 4, 5, 7, 8], 'cost': 260}, {'id': 21, 'elements': [6, 7, 8, 9], 'cost': 244}, {'id': 22, 'elements': [2, 7], 'cost': 98}, {'id': 23, 'elements': [3, 4, 5, 8], 'cost': 36}, {'id': 24, 'elements': [15, 16], 'cost': 138}, {'id': 25, 'elements': [1, 2, 4], 'cost': 51}, {'id': 26, 'elements': [10, 13, 15], 'cost': 192}, {'id': 27, 'elements': [5, 6, 7, 9], 'cost': 340}, {'id': 28, 'elements': [3, 5], 'cost': 158}, {'id': 29, 'elements': [7, 9], 'cost': 78}, {'id': 30, 'elements': [14, 15, 16], 'cost': 105}, {'id': 31, 'elements': [3, 4, 5, 7], 'cost': 20}, {'id': 32, 'elements': [9, 13, 15, 16], 'cost': 328}, {'id': 33, 'elements': [6, 8], 'cost': 148}, {'id': 34, 'elements': [11, 14, 15], 'cost': 156}, {'id': 35, 'elements': [6, 7, 8], 'cost': 288}, {'id': 36, 'elements': [10, 13], 'cost': 66}, {'id': 37, 'elements': [13, 15, 16], 'cost': 132}, {'id': 38, 'elements': [7, 9, 10, 12], 'cost': 68}, {'id': 39, 'elements': [4, 5, 7, 9], 'cost': 44}, {'id': 40, 'elements': [4, 6, 8, 9, 11], 'cost': 145}, {'id': 41, 'elements': [4, 6, 7, 8], 'cost': 272}, {'id': 42, 'elements': [12, 13, 16], 'cost': 66}, {'id': 43, 'elements': [4, 5, 6], 'cost': 87}, {'id': 44, 'elements': [12, 15], 'cost': 108}, {'id': 45, 'elements': [12, 13, 14, 16], 'cost': 364}, {'id': 46, 'elements': [12, 14], 'cost': 42}, {'id': 47, 'elements': [11, 14], 'cost': 174}, {'id': 48, 'elements': [13, 14], 'cost': 84}, {'id': 49, 'elements': [4, 5], 'cost': 198}, {'id': 50, 'elements': [13, 14, 15, 16], 'cost': 316}, {'id': 51, 'elements': [3, 4, 5], 'cost': 105}, {'id': 52, 'elements': [1, 2, 3, 4, 5], 'cost': 455}, {'id': 53, 'elements': [11, 12, 13, 14], 'cost': 224}, {'id': 54, 'elements': [3, 5, 6], 'cost': 108}, {'id': 55, 'elements': [8, 11, 13, 15], 'cost': 52}, {'id': 56, 'elements': [10, 11, 15], 'cost': 249}, {'id': 57, 'elements': [2, 3, 5, 6, 7], 'cost': 270}, {'id': 58, 'elements': [1, 2, 3, 4], 'cost': 268}, {'id': 59, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 66, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 67, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 68, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 69, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 70, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 71, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 72, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 73, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 74, 'elements': [16], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0036_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0036_bag_stats.png'}","[18, 25, 28, 33, 34, 38]",635.0,"{'num_elements': 16, 'num_sets': 74, 'sets': [{'id': 'S1', 'elements': [2, 3, 4, 5, 6], 'cost': 475}, {'id': 'S2', 'elements': [9, 12], 'cost': 88}, {'id': 'S3', 'elements': [13, 14, 16], 'cost': 240}, {'id': 'S4', 'elements': [1, 2, 3, 5], 'cost': 300}, {'id': 'S5', 'elements': [9, 11, 12, 13], 'cost': 184}, {'id': 'S6', 'elements': [2, 3, 4, 5], 'cost': 276}, {'id': 'S7', 'elements': [6, 7, 9], 'cost': 105}, {'id': 'S8', 'elements': [4, 8], 'cost': 114}, {'id': 'S9', 'elements': [10, 15], 'cost': 190}, {'id': 'S10', 'elements': [2, 3], 'cost': 54}, {'id': 'S11', 'elements': [2, 3, 5, 6], 'cost': 64}, {'id': 'S12', 'elements': [5, 7], 'cost': 156}, {'id': 'S13', 'elements': [9, 10, 11], 'cost': 195}, {'id': 'S14', 'elements': [5, 9, 12], 'cost': 147}, {'id': 'S15', 'elements': [11, 12, 13, 16], 'cost': 36}, {'id': 'S16', 'elements': [12, 14, 15, 16], 'cost': 24}, {'id': 'S17', 'elements': [4, 7], 'cost': 134}, {'id': 'S18', 'elements': [13, 16], 'cost': 54}, {'id': 'S19', 'elements': [8, 9, 10, 11], 'cost': 268}, {'id': 'S20', 'elements': [3, 4, 5, 7, 8], 'cost': 260}, {'id': 'S21', 'elements': [6, 7, 8, 9], 'cost': 244}, {'id': 'S22', 'elements': [2, 7], 'cost': 98}, {'id': 'S23', 'elements': [3, 4, 5, 8], 'cost': 36}, {'id': 'S24', 'elements': [15, 16], 'cost': 138}, {'id': 'S25', 'elements': [1, 2, 4], 'cost': 51}, {'id': 'S26', 'elements': [10, 13, 15], 'cost': 192}, {'id': 'S27', 'elements': [5, 6, 7, 9], 'cost': 340}, {'id': 'S28', 'elements': [3, 5], 'cost': 158}, {'id': 'S29', 'elements': [7, 9], 'cost': 78}, {'id': 'S30', 'elements': [14, 15, 16], 'cost': 105}, {'id': 'S31', 'elements': [3, 4, 5, 7], 'cost': 20}, {'id': 'S32', 'elements': [9, 13, 15, 16], 'cost': 328}, {'id': 'S33', 'elements': [6, 8], 'cost': 148}, {'id': 'S34', 'elements': [11, 14, 15], 'cost': 156}, {'id': 'S35', 'elements': [6, 7, 8], 'cost': 288}, {'id': 'S36', 'elements': [10, 13], 'cost': 66}, {'id': 'S37', 'elements': [13, 15, 16], 'cost': 132}, {'id': 'S38', 'elements': [7, 9, 10, 12], 'cost': 68}, {'id': 'S39', 'elements': [4, 5, 7, 9], 'cost': 44}, {'id': 'S40', 'elements': [4, 6, 8, 9, 11], 'cost': 145}, {'id': 'S41', 'elements': [4, 6, 7, 8], 'cost': 272}, {'id': 'S42', 'elements': [12, 13, 16], 'cost': 66}, {'id': 'S43', 'elements': [4, 5, 6], 'cost': 87}, {'id': 'S44', 'elements': [12, 15], 'cost': 108}, {'id': 'S45', 'elements': [12, 13, 14, 16], 'cost': 364}, {'id': 'S46', 'elements': [12, 14], 'cost': 42}, {'id': 'S47', 'elements': [11, 14], 'cost': 174}, {'id': 'S48', 'elements': [13, 14], 'cost': 84}, {'id': 'S49', 'elements': [4, 5], 'cost': 198}, {'id': 'S50', 'elements': [13, 14, 15, 16], 'cost': 316}, {'id': 'S51', 'elements': [3, 4, 5], 'cost': 105}, {'id': 'S52', 'elements': [1, 2, 3, 4, 5], 'cost': 455}, {'id': 'S53', 'elements': [11, 12, 13, 14], 'cost': 224}, {'id': 'S54', 'elements': [3, 5, 6], 'cost': 108}, {'id': 'S55', 'elements': [8, 11, 13, 15], 'cost': 52}, {'id': 'S56', 'elements': [10, 11, 15], 'cost': 249}, {'id': 'S57', 'elements': [2, 3, 5, 6, 7], 'cost': 270}, {'id': 'S58', 'elements': [1, 2, 3, 4], 'cost': 268}, {'id': 'S59', 'elements': [1], 'cost': 10000}, {'id': 'S60', 'elements': [2], 'cost': 10000}, {'id': 'S61', 'elements': [3], 'cost': 10000}, {'id': 'S62', 'elements': [4], 'cost': 10000}, {'id': 'S63', 'elements': [5], 'cost': 10000}, {'id': 'S64', 'elements': [6], 'cost': 10000}, {'id': 'S65', 'elements': [7], 'cost': 10000}, {'id': 'S66', 'elements': [8], 'cost': 10000}, {'id': 'S67', 'elements': [9], 'cost': 10000}, {'id': 'S68', 'elements': [10], 'cost': 10000}, {'id': 'S69', 'elements': [11], 'cost': 10000}, {'id': 'S70', 'elements': [12], 'cost': 10000}, {'id': 'S71', 'elements': [13], 'cost': 10000}, {'id': 'S72', 'elements': [14], 'cost': 10000}, {'id': 'S73', 'elements': [15], 'cost': 10000}, {'id': 'S74', 'elements': [16], 'cost': 10000}]}","['S18', 'S25', 'S28', 'S33', 'S34', 'S38']",37,csv,1
SPP,SPP,"There’s a scheduling puzzle at the depot: choose a handful of preplanned routes so that every neighborhood gets a single delivery and no neighborhood appears on more than one picked route. Each route carries a fuel price tag — total fuel is just the sum of those tags for the routes we pick, and the aim is to keep that sum as low as possible. The concrete route and cost details are listed below.

- **num_neighborhoods**: 13
- **num_routes**: 64

| route_id | fuel_cost | route_neighborhoods |
|---|---|---|
| S1 | 84 | J L M |
| S2 | 159 | A B C |
| S3 | 264 | K L M |
| S4 | 84 | C E |
| S5 | 54 | J L |
| S6 | 300 | I J K |
| S7 | 285 | G H I |
| S8 | 36 | C D F |
| S9 | 78 | F G H |
| S10 | 114 | K L |
| S11 | 160 | J K L M |
| S12 | 320 | I K L M |
| S13 | 255 | B E F |
| S14 | 92 | D E |
| S15 | 76 | E I |
| S16 | 96 | A B D |
| S17 | 267 | H J M |
| S18 | 152 | B C E F |
| S19 | 114 | A C |
| S20 | 48 | H M |
| S21 | 136 | I J K L |
| S22 | 93 | E F G |
| S23 | 84 | D G |
| S24 | 234 | G I J |
| S25 | 142 | F G |
| S26 | 240 | B D E |
| S27 | 90 | I J |
| S28 | 279 | J K M |
| S29 | 153 | F H I |
| S30 | 78 | G I |
| S31 | 106 | J K |
| S32 | 2 | I L |
| S33 | 316 | A B C E |
| S34 | 94 | H |
| S35 | 48 | K M |
| S36 | 43 | I |
| S37 | 256 | A C D E |
| S38 | 272 | A B C D |
| S39 | 4 | F |
| S40 | 75 | J K L |
| S41 | 396 | B C D E |
| S42 | 160 | C D |
| S43 | 16 | K |
| S44 | 228 | C E G |
| S45 | 132 | B C F |
| S46 | 189 | D E H |
| S47 | 124 | E F |
| S48 | 201 | I K L |
| S49 | 111 | H I J |
| S50 | 42 | D E G |
| S51 | 198 | G H |
| S52 | 10000 | A |
| S53 | 10000 | B |
| S54 | 10000 | C |
| S55 | 10000 | D |
| S56 | 10000 | E |
| S57 | 10000 | F |
| S58 | 10000 | G |
| S59 | 10000 | H |
| S60 | 10000 | I |
| S61 | 10000 | J |
| S62 | 10000 | K |
| S63 | 10000 | L |
| S64 | 10000 | M |

If you want to hand the chosen routes back in a tidy, machine-friendly way, just reply with a little JSON that looks like this:

{
  ""solution"": [""route_id"", ...]
}

Keep in mind this is just a sketch of the shape I expect: ""solution"" is the list of picked routes, and each ""route_id"" is a placeholder for the exact route identifier from the instance (use the identifier exactly as given). This isn't the actual answer — just the format to use.

Please don’t rename any identifiers or invent new labels; use the IDs exactly as they appear in the instance input — no renaming and no new labels.
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 13, 'num_sets': 64, 'density': 0.17908653846153846, 'sets': [{'id': 1, 'elements': [10, 12, 13], 'cost': 84}, {'id': 2, 'elements': [1, 2, 3], 'cost': 159}, {'id': 3, 'elements': [11, 12, 13], 'cost': 264}, {'id': 4, 'elements': [3, 5], 'cost': 84}, {'id': 5, 'elements': [10, 12], 'cost': 54}, {'id': 6, 'elements': [9, 10, 11], 'cost': 300}, {'id': 7, 'elements': [7, 8, 9], 'cost': 285}, {'id': 8, 'elements': [3, 4, 6], 'cost': 36}, {'id': 9, 'elements': [6, 7, 8], 'cost': 78}, {'id': 10, 'elements': [11, 12], 'cost': 114}, {'id': 11, 'elements': [10, 11, 12, 13], 'cost': 160}, {'id': 12, 'elements': [9, 11, 12, 13], 'cost': 320}, {'id': 13, 'elements': [2, 5, 6], 'cost': 255}, {'id': 14, 'elements': [4, 5], 'cost': 92}, {'id': 15, 'elements': [5, 9], 'cost': 76}, {'id': 16, 'elements': [1, 2, 4], 'cost': 96}, {'id': 17, 'elements': [8, 10, 13], 'cost': 267}, {'id': 18, 'elements': [2, 3, 5, 6], 'cost': 152}, {'id': 19, 'elements': [1, 3], 'cost': 114}, {'id': 20, 'elements': [8, 13], 'cost': 48}, {'id': 21, 'elements': [9, 10, 11, 12], 'cost': 136}, {'id': 22, 'elements': [5, 6, 7], 'cost': 93}, {'id': 23, 'elements': [4, 7], 'cost': 84}, {'id': 24, 'elements': [7, 9, 10], 'cost': 234}, {'id': 25, 'elements': [6, 7], 'cost': 142}, {'id': 26, 'elements': [2, 4, 5], 'cost': 240}, {'id': 27, 'elements': [9, 10], 'cost': 90}, {'id': 28, 'elements': [10, 11, 13], 'cost': 279}, {'id': 29, 'elements': [6, 8, 9], 'cost': 153}, {'id': 30, 'elements': [7, 9], 'cost': 78}, {'id': 31, 'elements': [10, 11], 'cost': 106}, {'id': 32, 'elements': [9, 12], 'cost': 2}, {'id': 33, 'elements': [1, 2, 3, 5], 'cost': 316}, {'id': 34, 'elements': [8], 'cost': 94}, {'id': 35, 'elements': [11, 13], 'cost': 48}, {'id': 36, 'elements': [9], 'cost': 43}, {'id': 37, 'elements': [1, 3, 4, 5], 'cost': 256}, {'id': 38, 'elements': [1, 2, 3, 4], 'cost': 272}, {'id': 39, 'elements': [6], 'cost': 4}, {'id': 40, 'elements': [10, 11, 12], 'cost': 75}, {'id': 41, 'elements': [2, 3, 4, 5], 'cost': 396}, {'id': 42, 'elements': [3, 4], 'cost': 160}, {'id': 43, 'elements': [11], 'cost': 16}, {'id': 44, 'elements': [3, 5, 7], 'cost': 228}, {'id': 45, 'elements': [2, 3, 6], 'cost': 132}, {'id': 46, 'elements': [4, 5, 8], 'cost': 189}, {'id': 47, 'elements': [5, 6], 'cost': 124}, {'id': 48, 'elements': [9, 11, 12], 'cost': 201}, {'id': 49, 'elements': [8, 9, 10], 'cost': 111}, {'id': 50, 'elements': [4, 5, 7], 'cost': 42}, {'id': 51, 'elements': [7, 8], 'cost': 198}, {'id': 52, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [13], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0037_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0037_bag_stats.png'}","[2, 20, 31, 32, 39, 50]",361.0,"{'num_elements': 13, 'num_sets': 64, 'sets': [{'id': 'S1', 'elements': ['J', 'L', 'M'], 'cost': 84}, {'id': 'S2', 'elements': ['A', 'B', 'C'], 'cost': 159}, {'id': 'S3', 'elements': ['K', 'L', 'M'], 'cost': 264}, {'id': 'S4', 'elements': ['C', 'E'], 'cost': 84}, {'id': 'S5', 'elements': ['J', 'L'], 'cost': 54}, {'id': 'S6', 'elements': ['I', 'J', 'K'], 'cost': 300}, {'id': 'S7', 'elements': ['G', 'H', 'I'], 'cost': 285}, {'id': 'S8', 'elements': ['C', 'D', 'F'], 'cost': 36}, {'id': 'S9', 'elements': ['F', 'G', 'H'], 'cost': 78}, {'id': 'S10', 'elements': ['K', 'L'], 'cost': 114}, {'id': 'S11', 'elements': ['J', 'K', 'L', 'M'], 'cost': 160}, {'id': 'S12', 'elements': ['I', 'K', 'L', 'M'], 'cost': 320}, {'id': 'S13', 'elements': ['B', 'E', 'F'], 'cost': 255}, {'id': 'S14', 'elements': ['D', 'E'], 'cost': 92}, {'id': 'S15', 'elements': ['E', 'I'], 'cost': 76}, {'id': 'S16', 'elements': ['A', 'B', 'D'], 'cost': 96}, {'id': 'S17', 'elements': ['H', 'J', 'M'], 'cost': 267}, {'id': 'S18', 'elements': ['B', 'C', 'E', 'F'], 'cost': 152}, {'id': 'S19', 'elements': ['A', 'C'], 'cost': 114}, {'id': 'S20', 'elements': ['H', 'M'], 'cost': 48}, {'id': 'S21', 'elements': ['I', 'J', 'K', 'L'], 'cost': 136}, {'id': 'S22', 'elements': ['E', 'F', 'G'], 'cost': 93}, {'id': 'S23', 'elements': ['D', 'G'], 'cost': 84}, {'id': 'S24', 'elements': ['G', 'I', 'J'], 'cost': 234}, {'id': 'S25', 'elements': ['F', 'G'], 'cost': 142}, {'id': 'S26', 'elements': ['B', 'D', 'E'], 'cost': 240}, {'id': 'S27', 'elements': ['I', 'J'], 'cost': 90}, {'id': 'S28', 'elements': ['J', 'K', 'M'], 'cost': 279}, {'id': 'S29', 'elements': ['F', 'H', 'I'], 'cost': 153}, {'id': 'S30', 'elements': ['G', 'I'], 'cost': 78}, {'id': 'S31', 'elements': ['J', 'K'], 'cost': 106}, {'id': 'S32', 'elements': ['I', 'L'], 'cost': 2}, {'id': 'S33', 'elements': ['A', 'B', 'C', 'E'], 'cost': 316}, {'id': 'S34', 'elements': ['H'], 'cost': 94}, {'id': 'S35', 'elements': ['K', 'M'], 'cost': 48}, {'id': 'S36', 'elements': ['I'], 'cost': 43}, {'id': 'S37', 'elements': ['A', 'C', 'D', 'E'], 'cost': 256}, {'id': 'S38', 'elements': ['A', 'B', 'C', 'D'], 'cost': 272}, {'id': 'S39', 'elements': ['F'], 'cost': 4}, {'id': 'S40', 'elements': ['J', 'K', 'L'], 'cost': 75}, {'id': 'S41', 'elements': ['B', 'C', 'D', 'E'], 'cost': 396}, {'id': 'S42', 'elements': ['C', 'D'], 'cost': 160}, {'id': 'S43', 'elements': ['K'], 'cost': 16}, {'id': 'S44', 'elements': ['C', 'E', 'G'], 'cost': 228}, {'id': 'S45', 'elements': ['B', 'C', 'F'], 'cost': 132}, {'id': 'S46', 'elements': ['D', 'E', 'H'], 'cost': 189}, {'id': 'S47', 'elements': ['E', 'F'], 'cost': 124}, {'id': 'S48', 'elements': ['I', 'K', 'L'], 'cost': 201}, {'id': 'S49', 'elements': ['H', 'I', 'J'], 'cost': 111}, {'id': 'S50', 'elements': ['D', 'E', 'G'], 'cost': 42}, {'id': 'S51', 'elements': ['G', 'H'], 'cost': 198}, {'id': 'S52', 'elements': ['A'], 'cost': 10000}, {'id': 'S53', 'elements': ['B'], 'cost': 10000}, {'id': 'S54', 'elements': ['C'], 'cost': 10000}, {'id': 'S55', 'elements': ['D'], 'cost': 10000}, {'id': 'S56', 'elements': ['E'], 'cost': 10000}, {'id': 'S57', 'elements': ['F'], 'cost': 10000}, {'id': 'S58', 'elements': ['G'], 'cost': 10000}, {'id': 'S59', 'elements': ['H'], 'cost': 10000}, {'id': 'S60', 'elements': ['I'], 'cost': 10000}, {'id': 'S61', 'elements': ['J'], 'cost': 10000}, {'id': 'S62', 'elements': ['K'], 'cost': 10000}, {'id': 'S63', 'elements': ['L'], 'cost': 10000}, {'id': 'S64', 'elements': ['M'], 'cost': 10000}]}","['S2', 'S20', 'S31', 'S32', 'S39', 'S50']",38,markdown_table,names
SPP,SPP,"Recently the library received a bunch of curated book bundles and the job is to choose which ones to shelve so every genre is represented, with each genre coming from exactly one bundle. Because some bundles share genres, the choices must be made so those overlaps don’t end up in the final set, and the practical goal is to minimize how much is spent by summing the prices of the chosen bundles. The complete list of bundles and prices is shown below.

{
  ""total_genres"": 13,
  ""total_bundles"": 48,
  ""sets"": [
    {
      ""bundle_id"": ""S1"",
      ""bundle_genres"": [
        9,
        10
      ],
      ""bundle_price"": 46
    },
    {
      ""bundle_id"": ""S2"",
      ""bundle_genres"": [
        8,
        9,
        12
      ],
      ""bundle_price"": 150
    },
    {
      ""bundle_id"": ""S3"",
      ""bundle_genres"": [
        6,
        9
      ],
      ""bundle_price"": 112
    },
    {
      ""bundle_id"": ""S4"",
      ""bundle_genres"": [
        7,
        8,
        9
      ],
      ""bundle_price"": 18
    },
    {
      ""bundle_id"": ""S5"",
      ""bundle_genres"": [
        0,
        1,
        2,
        3
      ],
      ""bundle_price"": 368
    },
    {
      ""bundle_id"": ""S6"",
      ""bundle_genres"": [
        3,
        4
      ],
      ""bundle_price"": 12
    },
    {
      ""bundle_id"": ""S7"",
      ""bundle_genres"": [
        1,
        2,
        4,
        5
      ],
      ""bundle_price"": 340
    },
    {
      ""bundle_id"": ""S8"",
      ""bundle_genres"": [
        6,
        7,
        8
      ],
      ""bundle_price"": 144
    },
    {
      ""bundle_id"": ""S9"",
      ""bundle_genres"": [
        4,
        5,
        6,
        7
      ],
      ""bundle_price"": 220
    },
    {
      ""bundle_id"": ""S10"",
      ""bundle_genres"": [
        9,
        11,
        12
      ],
      ""bundle_price"": 252
    },
    {
      ""bundle_id"": ""S11"",
      ""bundle_genres"": [
        1,
        4,
        5
      ],
      ""bundle_price"": 147
    },
    {
      ""bundle_id"": ""S12"",
      ""bundle_genres"": [
        8,
        10,
        11
      ],
      ""bundle_price"": 51
    },
    {
      ""bundle_id"": ""S13"",
      ""bundle_genres"": [
        1,
        2,
        3
      ],
      ""bundle_price"": 189
    },
    {
      ""bundle_id"": ""S14"",
      ""bundle_genres"": [
        7
      ],
      ""bundle_price"": 95
    },
    {
      ""bundle_id"": ""S15"",
      ""bundle_genres"": [
        10,
        11
      ],
      ""bundle_price"": 74
    },
    {
      ""bundle_id"": ""S16"",
      ""bundle_genres"": [
        3,
        4,
        5
      ],
      ""bundle_price"": 153
    },
    {
      ""bundle_id"": ""S17"",
      ""bundle_genres"": [
        2,
        3,
        5
      ],
      ""bundle_price"": 51
    },
    {
      ""bundle_id"": ""S18"",
      ""bundle_genres"": [
        0,
        1,
        2
      ],
      ""bundle_price"": 150
    },
    {
      ""bundle_id"": ""S19"",
      ""bundle_genres"": [
        9,
        12
      ],
      ""bundle_price"": 112
    },
    {
      ""bundle_id"": ""S20"",
      ""bundle_genres"": [
        6,
        7,
        9
      ],
      ""bundle_price"": 123
    },
    {
      ""bundle_id"": ""S21"",
      ""bundle_genres"": [
        2
      ],
      ""bundle_price"": 44
    },
    {
      ""bundle_id"": ""S22"",
      ""bundle_genres"": [
        3,
        4,
        5,
        6
      ],
      ""bundle_price"": 204
    },
    {
      ""bundle_id"": ""S23"",
      ""bundle_genres"": [
        4,
        6,
        7
      ],
      ""bundle_price"": 27
    },
    {
      ""bundle_id"": ""S24"",
      ""bundle_genres"": [
        7,
        9,
        10
      ],
      ""bundle_price"": 123
    },
    {
      ""bundle_id"": ""S25"",
      ""bundle_genres"": [
        8
      ],
      ""bundle_price"": 67
    },
    {
      ""bundle_id"": ""S26"",
      ""bundle_genres"": [
        8,
        9,
        10
      ],
      ""bundle_price"": 15
    },
    {
      ""bundle_id"": ""S27"",
      ""bundle_genres"": [
        0,
        2
      ],
      ""bundle_price"": 2
    },
    {
      ""bundle_id"": ""S28"",
      ""bundle_genres"": [
        4,
        5,
        7
      ],
      ""bundle_price"": 243
    },
    {
      ""bundle_id"": ""S29"",
      ""bundle_genres"": [
        6
      ],
      ""bundle_price"": 84
    },
    {
      ""bundle_id"": ""S30"",
      ""bundle_genres"": [
        10,
        12
      ],
      ""bundle_price"": 40
    },
    {
      ""bundle_id"": ""S31"",
      ""bundle_genres"": [
        4,
        5
      ],
      ""bundle_price"": 188
    },
    {
      ""bundle_id"": ""S32"",
      ""bundle_genres"": [
        7,
        10
      ],
      ""bundle_price"": 96
    },
    {
      ""bundle_id"": ""S33"",
      ""bundle_genres"": [
        6,
        9,
        11
      ],
      ""bundle_price"": 243
    },
    {
      ""bundle_id"": ""S34"",
      ""bundle_genres"": [
        5,
        6
      ],
      ""bundle_price"": 188
    },
    {
      ""bundle_id"": ""S35"",
      ""bundle_genres"": [
        5,
        6,
        7
      ],
      ""bundle_price"": 159
    },
    {
      ""bundle_id"": ""S36"",
      ""bundle_genres"": [
        0
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S37"",
      ""bundle_genres"": [
        1
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S38"",
      ""bundle_genres"": [
        2
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S39"",
      ""bundle_genres"": [
        3
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S40"",
      ""bundle_genres"": [
        4
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S41"",
      ""bundle_genres"": [
        5
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S42"",
      ""bundle_genres"": [
        6
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S43"",
      ""bundle_genres"": [
        7
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S44"",
      ""bundle_genres"": [
        8
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S45"",
      ""bundle_genres"": [
        9
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S46"",
      ""bundle_genres"": [
        10
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S47"",
      ""bundle_genres"": [
        11
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S48"",
      ""bundle_genres"": [
        12
      ],
      ""bundle_price"": 10000
    }
  ]
}

If you want to tell me which bundles to shelve, a tiny JSON snippet like this works great — keeps things neat and machine-friendly but still simple to read:

{
  ""solution"": [""bundle_id"", ...]
}

""solution"" is a list of the bundle identifiers you choose to shelve so every genre is represented exactly once. Each item in that array should be the identifier for a bundle from the list above. Think of it like filling out a short form: list the bundle IDs you pick, and that's it.

This JSON is just a sketch of the shape I expect, not the actual answer — put the real bundle IDs from the instance into the array when you're ready.

Also, please make sure all identifiers are used exactly as they appear in the instance input — no renaming and no new labels.

for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 13, 'num_sets': 48, 'density': 0.16666666666666666, 'sets': [{'id': 1, 'elements': [10, 11], 'cost': 46}, {'id': 2, 'elements': [9, 10, 13], 'cost': 150}, {'id': 3, 'elements': [7, 10], 'cost': 112}, {'id': 4, 'elements': [8, 9, 10], 'cost': 18}, {'id': 5, 'elements': [1, 2, 3, 4], 'cost': 368}, {'id': 6, 'elements': [4, 5], 'cost': 12}, {'id': 7, 'elements': [2, 3, 5, 6], 'cost': 340}, {'id': 8, 'elements': [7, 8, 9], 'cost': 144}, {'id': 9, 'elements': [5, 6, 7, 8], 'cost': 220}, {'id': 10, 'elements': [10, 12, 13], 'cost': 252}, {'id': 11, 'elements': [2, 5, 6], 'cost': 147}, {'id': 12, 'elements': [9, 11, 12], 'cost': 51}, {'id': 13, 'elements': [2, 3, 4], 'cost': 189}, {'id': 14, 'elements': [8], 'cost': 95}, {'id': 15, 'elements': [11, 12], 'cost': 74}, {'id': 16, 'elements': [4, 5, 6], 'cost': 153}, {'id': 17, 'elements': [3, 4, 6], 'cost': 51}, {'id': 18, 'elements': [1, 2, 3], 'cost': 150}, {'id': 19, 'elements': [10, 13], 'cost': 112}, {'id': 20, 'elements': [7, 8, 10], 'cost': 123}, {'id': 21, 'elements': [3], 'cost': 44}, {'id': 22, 'elements': [4, 5, 6, 7], 'cost': 204}, {'id': 23, 'elements': [5, 7, 8], 'cost': 27}, {'id': 24, 'elements': [8, 10, 11], 'cost': 123}, {'id': 25, 'elements': [9], 'cost': 67}, {'id': 26, 'elements': [9, 10, 11], 'cost': 15}, {'id': 27, 'elements': [1, 3], 'cost': 2}, {'id': 28, 'elements': [5, 6, 8], 'cost': 243}, {'id': 29, 'elements': [7], 'cost': 84}, {'id': 30, 'elements': [11, 13], 'cost': 40}, {'id': 31, 'elements': [5, 6], 'cost': 188}, {'id': 32, 'elements': [8, 11], 'cost': 96}, {'id': 33, 'elements': [7, 10, 12], 'cost': 243}, {'id': 34, 'elements': [6, 7], 'cost': 188}, {'id': 35, 'elements': [6, 7, 8], 'cost': 159}, {'id': 36, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 37, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 38, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 39, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 40, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 41, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 42, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 43, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 44, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 45, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 46, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 47, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [13], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0038_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0038_bag_stats.png'}","[6, 12, 18, 19, 35]",484.0,"{'num_elements': 13, 'num_sets': 48, 'sets': [{'id': 'S1', 'elements': [9, 10], 'cost': 46}, {'id': 'S2', 'elements': [8, 9, 12], 'cost': 150}, {'id': 'S3', 'elements': [6, 9], 'cost': 112}, {'id': 'S4', 'elements': [7, 8, 9], 'cost': 18}, {'id': 'S5', 'elements': [0, 1, 2, 3], 'cost': 368}, {'id': 'S6', 'elements': [3, 4], 'cost': 12}, {'id': 'S7', 'elements': [1, 2, 4, 5], 'cost': 340}, {'id': 'S8', 'elements': [6, 7, 8], 'cost': 144}, {'id': 'S9', 'elements': [4, 5, 6, 7], 'cost': 220}, {'id': 'S10', 'elements': [9, 11, 12], 'cost': 252}, {'id': 'S11', 'elements': [1, 4, 5], 'cost': 147}, {'id': 'S12', 'elements': [8, 10, 11], 'cost': 51}, {'id': 'S13', 'elements': [1, 2, 3], 'cost': 189}, {'id': 'S14', 'elements': [7], 'cost': 95}, {'id': 'S15', 'elements': [10, 11], 'cost': 74}, {'id': 'S16', 'elements': [3, 4, 5], 'cost': 153}, {'id': 'S17', 'elements': [2, 3, 5], 'cost': 51}, {'id': 'S18', 'elements': [0, 1, 2], 'cost': 150}, {'id': 'S19', 'elements': [9, 12], 'cost': 112}, {'id': 'S20', 'elements': [6, 7, 9], 'cost': 123}, {'id': 'S21', 'elements': [2], 'cost': 44}, {'id': 'S22', 'elements': [3, 4, 5, 6], 'cost': 204}, {'id': 'S23', 'elements': [4, 6, 7], 'cost': 27}, {'id': 'S24', 'elements': [7, 9, 10], 'cost': 123}, {'id': 'S25', 'elements': [8], 'cost': 67}, {'id': 'S26', 'elements': [8, 9, 10], 'cost': 15}, {'id': 'S27', 'elements': [0, 2], 'cost': 2}, {'id': 'S28', 'elements': [4, 5, 7], 'cost': 243}, {'id': 'S29', 'elements': [6], 'cost': 84}, {'id': 'S30', 'elements': [10, 12], 'cost': 40}, {'id': 'S31', 'elements': [4, 5], 'cost': 188}, {'id': 'S32', 'elements': [7, 10], 'cost': 96}, {'id': 'S33', 'elements': [6, 9, 11], 'cost': 243}, {'id': 'S34', 'elements': [5, 6], 'cost': 188}, {'id': 'S35', 'elements': [5, 6, 7], 'cost': 159}, {'id': 'S36', 'elements': [0], 'cost': 10000}, {'id': 'S37', 'elements': [1], 'cost': 10000}, {'id': 'S38', 'elements': [2], 'cost': 10000}, {'id': 'S39', 'elements': [3], 'cost': 10000}, {'id': 'S40', 'elements': [4], 'cost': 10000}, {'id': 'S41', 'elements': [5], 'cost': 10000}, {'id': 'S42', 'elements': [6], 'cost': 10000}, {'id': 'S43', 'elements': [7], 'cost': 10000}, {'id': 'S44', 'elements': [8], 'cost': 10000}, {'id': 'S45', 'elements': [9], 'cost': 10000}, {'id': 'S46', 'elements': [10], 'cost': 10000}, {'id': 'S47', 'elements': [11], 'cost': 10000}, {'id': 'S48', 'elements': [12], 'cost': 10000}]}","['S6', 'S12', 'S18', 'S19', 'S35']",39,json,0
SPP,SPP,"I manage a mid‑size building and need to pick which cleaning packages to hire so every room gets cleaned exactly once. The idea is to choose a set of packages that together cover all rooms without any room being in two different packages or being skipped, and then add up the labor fees of the chosen packages — the plan with the lowest total fee is the one to go with. The specific rooms, package options, and fees are shown below.

# total_rooms=15
# num_packages=64
package_id,labor_fee,package_rooms
S1,116,3 4
S2,106,14 15
S3,170,3 6
S4,81,6 9 12
S5,297,10 12 14
S6,40,7 9
S7,189,7 9 10
S8,6,1 2 3
S9,104,11 13 14 15
S10,232,2 3 4 5
S11,63,11 12 14
S12,288,1 2 3 5
S13,114,1 3
S14,98,1 2
S15,171,10 13 14
S16,258,1 3 4
S17,69,12 14 15
S18,30,6 7
S19,44,11 12 13 15
S20,66,12 13 14
S21,12,4 5 6
S22,102,13 14 15
S23,255,6 7 11
S24,240,1 2 4
S25,122,7 10
S26,132,8 10 11
S27,258,3 5 6
S28,56,12 13 14 15
S29,176,11 12 13 14
S30,324,8 10 12 13
S31,384,10 11 13 14
S32,120,1 2 4 6
S33,136,10 12
S34,156,13 15
S35,96,5 7 9 10
S36,72,2 4 5
S37,135,7 8 10
S38,84,5 8 9 11
S39,16,2 4
S40,141,11 13 14
S41,232,8 9 11 12
S42,12,11 13
S43,208,6 7 9 10
S44,92,1 3 4 5
S45,126,12 14
S46,258,6 8 11
S47,42,5 7
S48,51,3 4 6
S49,144,6 7 8
S50,10000,1
S51,10000,2
S52,10000,3
S53,10000,4
S54,10000,5
S55,10000,6
S56,10000,7
S57,10000,8
S58,10000,9
S59,10000,10
S60,10000,11
S61,10000,12
S62,10000,13
S63,10000,14
S64,10000,15

If you want to return the chosen cleaning packages, just use this simple JSON layout when you reply:

{
  ""solution"": [""package_id"", ...]
}

""solution"" should hold a list of the package identifiers you picked — each entry is the exact id of a cleaning package to hire. This JSON is just a sketch of the shape I expect, not the final answer; when you give the real plan, replace the placeholder package_id items with the actual ids from the instance. Please don’t rename or invent ids.

For example: Valid identifiers look like plain numbers such as ""1"" or ""23"", single capital letters like ""A"" or ""B"", or a capital letter followed by digits like ""A1"" or ""X7"".","{'problem_type': 'SPP', 'num_elements': 15, 'num_sets': 64, 'density': 0.16875, 'sets': [{'id': 1, 'elements': [3, 4], 'cost': 116}, {'id': 2, 'elements': [14, 15], 'cost': 106}, {'id': 3, 'elements': [3, 6], 'cost': 170}, {'id': 4, 'elements': [6, 9, 12], 'cost': 81}, {'id': 5, 'elements': [10, 12, 14], 'cost': 297}, {'id': 6, 'elements': [7, 9], 'cost': 40}, {'id': 7, 'elements': [7, 9, 10], 'cost': 189}, {'id': 8, 'elements': [1, 2, 3], 'cost': 6}, {'id': 9, 'elements': [11, 13, 14, 15], 'cost': 104}, {'id': 10, 'elements': [2, 3, 4, 5], 'cost': 232}, {'id': 11, 'elements': [11, 12, 14], 'cost': 63}, {'id': 12, 'elements': [1, 2, 3, 5], 'cost': 288}, {'id': 13, 'elements': [1, 3], 'cost': 114}, {'id': 14, 'elements': [1, 2], 'cost': 98}, {'id': 15, 'elements': [10, 13, 14], 'cost': 171}, {'id': 16, 'elements': [1, 3, 4], 'cost': 258}, {'id': 17, 'elements': [12, 14, 15], 'cost': 69}, {'id': 18, 'elements': [6, 7], 'cost': 30}, {'id': 19, 'elements': [11, 12, 13, 15], 'cost': 44}, {'id': 20, 'elements': [12, 13, 14], 'cost': 66}, {'id': 21, 'elements': [4, 5, 6], 'cost': 12}, {'id': 22, 'elements': [13, 14, 15], 'cost': 102}, {'id': 23, 'elements': [6, 7, 11], 'cost': 255}, {'id': 24, 'elements': [1, 2, 4], 'cost': 240}, {'id': 25, 'elements': [7, 10], 'cost': 122}, {'id': 26, 'elements': [8, 10, 11], 'cost': 132}, {'id': 27, 'elements': [3, 5, 6], 'cost': 258}, {'id': 28, 'elements': [12, 13, 14, 15], 'cost': 56}, {'id': 29, 'elements': [11, 12, 13, 14], 'cost': 176}, {'id': 30, 'elements': [8, 10, 12, 13], 'cost': 324}, {'id': 31, 'elements': [10, 11, 13, 14], 'cost': 384}, {'id': 32, 'elements': [1, 2, 4, 6], 'cost': 120}, {'id': 33, 'elements': [10, 12], 'cost': 136}, {'id': 34, 'elements': [13, 15], 'cost': 156}, {'id': 35, 'elements': [5, 7, 9, 10], 'cost': 96}, {'id': 36, 'elements': [2, 4, 5], 'cost': 72}, {'id': 37, 'elements': [7, 8, 10], 'cost': 135}, {'id': 38, 'elements': [5, 8, 9, 11], 'cost': 84}, {'id': 39, 'elements': [2, 4], 'cost': 16}, {'id': 40, 'elements': [11, 13, 14], 'cost': 141}, {'id': 41, 'elements': [8, 9, 11, 12], 'cost': 232}, {'id': 42, 'elements': [11, 13], 'cost': 12}, {'id': 43, 'elements': [6, 7, 9, 10], 'cost': 208}, {'id': 44, 'elements': [1, 3, 4, 5], 'cost': 92}, {'id': 45, 'elements': [12, 14], 'cost': 126}, {'id': 46, 'elements': [6, 8, 11], 'cost': 258}, {'id': 47, 'elements': [5, 7], 'cost': 42}, {'id': 48, 'elements': [3, 4, 6], 'cost': 51}, {'id': 49, 'elements': [6, 7, 8], 'cost': 144}, {'id': 50, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [15], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0039_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0039_bag_stats.png'}","[6, 8, 21, 26, 28]",246.0,"{'num_elements': 15, 'num_sets': 64, 'sets': [{'id': 'S1', 'elements': [3, 4], 'cost': 116}, {'id': 'S2', 'elements': [14, 15], 'cost': 106}, {'id': 'S3', 'elements': [3, 6], 'cost': 170}, {'id': 'S4', 'elements': [6, 9, 12], 'cost': 81}, {'id': 'S5', 'elements': [10, 12, 14], 'cost': 297}, {'id': 'S6', 'elements': [7, 9], 'cost': 40}, {'id': 'S7', 'elements': [7, 9, 10], 'cost': 189}, {'id': 'S8', 'elements': [1, 2, 3], 'cost': 6}, {'id': 'S9', 'elements': [11, 13, 14, 15], 'cost': 104}, {'id': 'S10', 'elements': [2, 3, 4, 5], 'cost': 232}, {'id': 'S11', 'elements': [11, 12, 14], 'cost': 63}, {'id': 'S12', 'elements': [1, 2, 3, 5], 'cost': 288}, {'id': 'S13', 'elements': [1, 3], 'cost': 114}, {'id': 'S14', 'elements': [1, 2], 'cost': 98}, {'id': 'S15', 'elements': [10, 13, 14], 'cost': 171}, {'id': 'S16', 'elements': [1, 3, 4], 'cost': 258}, {'id': 'S17', 'elements': [12, 14, 15], 'cost': 69}, {'id': 'S18', 'elements': [6, 7], 'cost': 30}, {'id': 'S19', 'elements': [11, 12, 13, 15], 'cost': 44}, {'id': 'S20', 'elements': [12, 13, 14], 'cost': 66}, {'id': 'S21', 'elements': [4, 5, 6], 'cost': 12}, {'id': 'S22', 'elements': [13, 14, 15], 'cost': 102}, {'id': 'S23', 'elements': [6, 7, 11], 'cost': 255}, {'id': 'S24', 'elements': [1, 2, 4], 'cost': 240}, {'id': 'S25', 'elements': [7, 10], 'cost': 122}, {'id': 'S26', 'elements': [8, 10, 11], 'cost': 132}, {'id': 'S27', 'elements': [3, 5, 6], 'cost': 258}, {'id': 'S28', 'elements': [12, 13, 14, 15], 'cost': 56}, {'id': 'S29', 'elements': [11, 12, 13, 14], 'cost': 176}, {'id': 'S30', 'elements': [8, 10, 12, 13], 'cost': 324}, {'id': 'S31', 'elements': [10, 11, 13, 14], 'cost': 384}, {'id': 'S32', 'elements': [1, 2, 4, 6], 'cost': 120}, {'id': 'S33', 'elements': [10, 12], 'cost': 136}, {'id': 'S34', 'elements': [13, 15], 'cost': 156}, {'id': 'S35', 'elements': [5, 7, 9, 10], 'cost': 96}, {'id': 'S36', 'elements': [2, 4, 5], 'cost': 72}, {'id': 'S37', 'elements': [7, 8, 10], 'cost': 135}, {'id': 'S38', 'elements': [5, 8, 9, 11], 'cost': 84}, {'id': 'S39', 'elements': [2, 4], 'cost': 16}, {'id': 'S40', 'elements': [11, 13, 14], 'cost': 141}, {'id': 'S41', 'elements': [8, 9, 11, 12], 'cost': 232}, {'id': 'S42', 'elements': [11, 13], 'cost': 12}, {'id': 'S43', 'elements': [6, 7, 9, 10], 'cost': 208}, {'id': 'S44', 'elements': [1, 3, 4, 5], 'cost': 92}, {'id': 'S45', 'elements': [12, 14], 'cost': 126}, {'id': 'S46', 'elements': [6, 8, 11], 'cost': 258}, {'id': 'S47', 'elements': [5, 7], 'cost': 42}, {'id': 'S48', 'elements': [3, 4, 6], 'cost': 51}, {'id': 'S49', 'elements': [6, 7, 8], 'cost': 144}, {'id': 'S50', 'elements': [1], 'cost': 10000}, {'id': 'S51', 'elements': [2], 'cost': 10000}, {'id': 'S52', 'elements': [3], 'cost': 10000}, {'id': 'S53', 'elements': [4], 'cost': 10000}, {'id': 'S54', 'elements': [5], 'cost': 10000}, {'id': 'S55', 'elements': [6], 'cost': 10000}, {'id': 'S56', 'elements': [7], 'cost': 10000}, {'id': 'S57', 'elements': [8], 'cost': 10000}, {'id': 'S58', 'elements': [9], 'cost': 10000}, {'id': 'S59', 'elements': [10], 'cost': 10000}, {'id': 'S60', 'elements': [11], 'cost': 10000}, {'id': 'S61', 'elements': [12], 'cost': 10000}, {'id': 'S62', 'elements': [13], 'cost': 10000}, {'id': 'S63', 'elements': [14], 'cost': 10000}, {'id': 'S64', 'elements': [15], 'cost': 10000}]}","['S6', 'S8', 'S21', 'S26', 'S28']",40,csv,1
SPP,SPP,"We put together a shortlist of vendor modules and needed to decide which ones to bring into the product. The plan was to make sure each product feature ends up covered by a single chosen module, with no overlap between selected vendors and nothing left unhandled. To compare options, simply total the license costs of the modules you pick; the smaller that sum, the more attractive the combination. The concrete options and feature list are shown below.

- **num_features_total**: 17
- **num_vendor_modules**: 87

| module_id | license_cost | features_handled |
|---|---|---|
| S1 | 84 | M N |
| S2 | 176 | I J L M |
| S3 | 296 | K L M N |
| S4 | 15 | F G H I J |
| S5 | 204 | B C D |
| S6 | 297 | F H I |
| S7 | 87 | D E G |
| S8 | 164 | J K L M |
| S9 | 249 | K L M |
| S10 | 50 | A C |
| S11 | 6 | O P Q |
| S12 | 294 | D E H |
| S13 | 243 | J L N |
| S14 | 108 | D F H |
| S15 | 51 | E F H |
| S16 | 144 | A B D |
| S17 | 168 | K M N O |
| S18 | 36 | A B C D |
| S19 | 230 | C D E G H |
| S20 | 24 | K M N |
| S21 | 81 | J M N |
| S22 | 24 | B E F |
| S23 | 36 | J K |
| S24 | 198 | J K L |
| S25 | 104 | M N O P |
| S26 | 192 | D F G H |
| S27 | 243 | B D F |
| S28 | 388 | I J K M |
| S29 | 156 | N O P |
| S30 | 192 | A F |
| S31 | 380 | E F G H I |
| S32 | 132 | J L M O |
| S33 | 48 | A B |
| S34 | 128 | N O P Q |
| S35 | 160 | E F |
| S36 | 78 | C D |
| S37 | 165 | D F G |
| S38 | 64 | H J L N |
| S39 | 81 | I L M |
| S40 | 150 | B C |
| S41 | 80 | H I J K |
| S42 | 128 | J N |
| S43 | 18 | N O Q |
| S44 | 344 | G H I J |
| S45 | 252 | B F G |
| S46 | 60 | N O |
| S47 | 280 | A B C E |
| S48 | 84 | L M |
| S49 | 396 | K L M P |
| S50 | 92 | H K |
| S51 | 260 | L M P Q |
| S52 | 90 | C E F G H |
| S53 | 350 | F H I J K |
| S54 | 168 | E F G H |
| S55 | 80 | L N O Q |
| S56 | 125 | G I J K L |
| S57 | 10 | A B C E F |
| S58 | 248 | H I J L |
| S59 | 132 | A B C F |
| S60 | 180 | L M N O |
| S61 | 165 | L N O P Q |
| S62 | 196 | A B D E |
| S63 | 364 | F G H I |
| S64 | 120 | P Q |
| S65 | 99 | M N O |
| S66 | 30 | K L O |
| S67 | 192 | O P |
| S68 | 352 | C E F G |
| S69 | 156 | C H |
| S70 | 275 | J K L M N |
| S71 | 10000 | A |
| S72 | 10000 | B |
| S73 | 10000 | C |
| S74 | 10000 | D |
| S75 | 10000 | E |
| S76 | 10000 | F |
| S77 | 10000 | G |
| S78 | 10000 | H |
| S79 | 10000 | I |
| S80 | 10000 | J |
| S81 | 10000 | K |
| S82 | 10000 | L |
| S83 | 10000 | M |
| S84 | 10000 | N |
| S85 | 10000 | O |
| S86 | 10000 | P |
| S87 | 10000 | Q |

When you send your pick, just return a tiny JSON object in this shape — super simple, like filling out a short form:

{
  ""solution"": [""module_id"", ...]
}

""solution"" is the list of vendor modules you choose (one entry per feature, no overlaps). The placeholder module_id shows where each chosen module's identifier goes; the JSON above is just a sketch of the shape to follow, not the actual answer — replace the placeholder with the real IDs from the instance.

Quick reminder: all identifiers must be used exactly as they appear in the instance input — no renaming and no new labels.

Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'problem_type': 'SPP', 'num_elements': 17, 'num_sets': 87, 'density': 0.17173766058147397, 'sets': [{'id': 1, 'elements': [13, 14], 'cost': 84}, {'id': 2, 'elements': [9, 10, 12, 13], 'cost': 176}, {'id': 3, 'elements': [11, 12, 13, 14], 'cost': 296}, {'id': 4, 'elements': [6, 7, 8, 9, 10], 'cost': 15}, {'id': 5, 'elements': [2, 3, 4], 'cost': 204}, {'id': 6, 'elements': [6, 8, 9], 'cost': 297}, {'id': 7, 'elements': [4, 5, 7], 'cost': 87}, {'id': 8, 'elements': [10, 11, 12, 13], 'cost': 164}, {'id': 9, 'elements': [11, 12, 13], 'cost': 249}, {'id': 10, 'elements': [1, 3], 'cost': 50}, {'id': 11, 'elements': [15, 16, 17], 'cost': 6}, {'id': 12, 'elements': [4, 5, 8], 'cost': 294}, {'id': 13, 'elements': [10, 12, 14], 'cost': 243}, {'id': 14, 'elements': [4, 6, 8], 'cost': 108}, {'id': 15, 'elements': [5, 6, 8], 'cost': 51}, {'id': 16, 'elements': [1, 2, 4], 'cost': 144}, {'id': 17, 'elements': [11, 13, 14, 15], 'cost': 168}, {'id': 18, 'elements': [1, 2, 3, 4], 'cost': 36}, {'id': 19, 'elements': [3, 4, 5, 7, 8], 'cost': 230}, {'id': 20, 'elements': [11, 13, 14], 'cost': 24}, {'id': 21, 'elements': [10, 13, 14], 'cost': 81}, {'id': 22, 'elements': [2, 5, 6], 'cost': 24}, {'id': 23, 'elements': [10, 11], 'cost': 36}, {'id': 24, 'elements': [10, 11, 12], 'cost': 198}, {'id': 25, 'elements': [13, 14, 15, 16], 'cost': 104}, {'id': 26, 'elements': [4, 6, 7, 8], 'cost': 192}, {'id': 27, 'elements': [2, 4, 6], 'cost': 243}, {'id': 28, 'elements': [9, 10, 11, 13], 'cost': 388}, {'id': 29, 'elements': [14, 15, 16], 'cost': 156}, {'id': 30, 'elements': [1, 6], 'cost': 192}, {'id': 31, 'elements': [5, 6, 7, 8, 9], 'cost': 380}, {'id': 32, 'elements': [10, 12, 13, 15], 'cost': 132}, {'id': 33, 'elements': [1, 2], 'cost': 48}, {'id': 34, 'elements': [14, 15, 16, 17], 'cost': 128}, {'id': 35, 'elements': [5, 6], 'cost': 160}, {'id': 36, 'elements': [3, 4], 'cost': 78}, {'id': 37, 'elements': [4, 6, 7], 'cost': 165}, {'id': 38, 'elements': [8, 10, 12, 14], 'cost': 64}, {'id': 39, 'elements': [9, 12, 13], 'cost': 81}, {'id': 40, 'elements': [2, 3], 'cost': 150}, {'id': 41, 'elements': [8, 9, 10, 11], 'cost': 80}, {'id': 42, 'elements': [10, 14], 'cost': 128}, {'id': 43, 'elements': [14, 15, 17], 'cost': 18}, {'id': 44, 'elements': [7, 8, 9, 10], 'cost': 344}, {'id': 45, 'elements': [2, 6, 7], 'cost': 252}, {'id': 46, 'elements': [14, 15], 'cost': 60}, {'id': 47, 'elements': [1, 2, 3, 5], 'cost': 280}, {'id': 48, 'elements': [12, 13], 'cost': 84}, {'id': 49, 'elements': [11, 12, 13, 16], 'cost': 396}, {'id': 50, 'elements': [8, 11], 'cost': 92}, {'id': 51, 'elements': [12, 13, 16, 17], 'cost': 260}, {'id': 52, 'elements': [3, 5, 6, 7, 8], 'cost': 90}, {'id': 53, 'elements': [6, 8, 9, 10, 11], 'cost': 350}, {'id': 54, 'elements': [5, 6, 7, 8], 'cost': 168}, {'id': 55, 'elements': [12, 14, 15, 17], 'cost': 80}, {'id': 56, 'elements': [7, 9, 10, 11, 12], 'cost': 125}, {'id': 57, 'elements': [1, 2, 3, 5, 6], 'cost': 10}, {'id': 58, 'elements': [8, 9, 10, 12], 'cost': 248}, {'id': 59, 'elements': [1, 2, 3, 6], 'cost': 132}, {'id': 60, 'elements': [12, 13, 14, 15], 'cost': 180}, {'id': 61, 'elements': [12, 14, 15, 16, 17], 'cost': 165}, {'id': 62, 'elements': [1, 2, 4, 5], 'cost': 196}, {'id': 63, 'elements': [6, 7, 8, 9], 'cost': 364}, {'id': 64, 'elements': [16, 17], 'cost': 120}, {'id': 65, 'elements': [13, 14, 15], 'cost': 99}, {'id': 66, 'elements': [11, 12, 15], 'cost': 30}, {'id': 67, 'elements': [15, 16], 'cost': 192}, {'id': 68, 'elements': [3, 5, 6, 7], 'cost': 352}, {'id': 69, 'elements': [3, 8], 'cost': 156}, {'id': 70, 'elements': [10, 11, 12, 13, 14], 'cost': 275}, {'id': 71, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 72, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 73, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 74, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 75, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 76, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 77, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 78, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 79, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 80, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 81, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 82, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 83, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 84, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 85, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 86, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 87, 'elements': [17], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0040_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0040_bag_stats.png'}","[1, 11, 15, 18, 56]",302.0,"{'num_elements': 17, 'num_sets': 87, 'sets': [{'id': 'S1', 'elements': ['M', 'N'], 'cost': 84}, {'id': 'S2', 'elements': ['I', 'J', 'L', 'M'], 'cost': 176}, {'id': 'S3', 'elements': ['K', 'L', 'M', 'N'], 'cost': 296}, {'id': 'S4', 'elements': ['F', 'G', 'H', 'I', 'J'], 'cost': 15}, {'id': 'S5', 'elements': ['B', 'C', 'D'], 'cost': 204}, {'id': 'S6', 'elements': ['F', 'H', 'I'], 'cost': 297}, {'id': 'S7', 'elements': ['D', 'E', 'G'], 'cost': 87}, {'id': 'S8', 'elements': ['J', 'K', 'L', 'M'], 'cost': 164}, {'id': 'S9', 'elements': ['K', 'L', 'M'], 'cost': 249}, {'id': 'S10', 'elements': ['A', 'C'], 'cost': 50}, {'id': 'S11', 'elements': ['O', 'P', 'Q'], 'cost': 6}, {'id': 'S12', 'elements': ['D', 'E', 'H'], 'cost': 294}, {'id': 'S13', 'elements': ['J', 'L', 'N'], 'cost': 243}, {'id': 'S14', 'elements': ['D', 'F', 'H'], 'cost': 108}, {'id': 'S15', 'elements': ['E', 'F', 'H'], 'cost': 51}, {'id': 'S16', 'elements': ['A', 'B', 'D'], 'cost': 144}, {'id': 'S17', 'elements': ['K', 'M', 'N', 'O'], 'cost': 168}, {'id': 'S18', 'elements': ['A', 'B', 'C', 'D'], 'cost': 36}, {'id': 'S19', 'elements': ['C', 'D', 'E', 'G', 'H'], 'cost': 230}, {'id': 'S20', 'elements': ['K', 'M', 'N'], 'cost': 24}, {'id': 'S21', 'elements': ['J', 'M', 'N'], 'cost': 81}, {'id': 'S22', 'elements': ['B', 'E', 'F'], 'cost': 24}, {'id': 'S23', 'elements': ['J', 'K'], 'cost': 36}, {'id': 'S24', 'elements': ['J', 'K', 'L'], 'cost': 198}, {'id': 'S25', 'elements': ['M', 'N', 'O', 'P'], 'cost': 104}, {'id': 'S26', 'elements': ['D', 'F', 'G', 'H'], 'cost': 192}, {'id': 'S27', 'elements': ['B', 'D', 'F'], 'cost': 243}, {'id': 'S28', 'elements': ['I', 'J', 'K', 'M'], 'cost': 388}, {'id': 'S29', 'elements': ['N', 'O', 'P'], 'cost': 156}, {'id': 'S30', 'elements': ['A', 'F'], 'cost': 192}, {'id': 'S31', 'elements': ['E', 'F', 'G', 'H', 'I'], 'cost': 380}, {'id': 'S32', 'elements': ['J', 'L', 'M', 'O'], 'cost': 132}, {'id': 'S33', 'elements': ['A', 'B'], 'cost': 48}, {'id': 'S34', 'elements': ['N', 'O', 'P', 'Q'], 'cost': 128}, {'id': 'S35', 'elements': ['E', 'F'], 'cost': 160}, {'id': 'S36', 'elements': ['C', 'D'], 'cost': 78}, {'id': 'S37', 'elements': ['D', 'F', 'G'], 'cost': 165}, {'id': 'S38', 'elements': ['H', 'J', 'L', 'N'], 'cost': 64}, {'id': 'S39', 'elements': ['I', 'L', 'M'], 'cost': 81}, {'id': 'S40', 'elements': ['B', 'C'], 'cost': 150}, {'id': 'S41', 'elements': ['H', 'I', 'J', 'K'], 'cost': 80}, {'id': 'S42', 'elements': ['J', 'N'], 'cost': 128}, {'id': 'S43', 'elements': ['N', 'O', 'Q'], 'cost': 18}, {'id': 'S44', 'elements': ['G', 'H', 'I', 'J'], 'cost': 344}, {'id': 'S45', 'elements': ['B', 'F', 'G'], 'cost': 252}, {'id': 'S46', 'elements': ['N', 'O'], 'cost': 60}, {'id': 'S47', 'elements': ['A', 'B', 'C', 'E'], 'cost': 280}, {'id': 'S48', 'elements': ['L', 'M'], 'cost': 84}, {'id': 'S49', 'elements': ['K', 'L', 'M', 'P'], 'cost': 396}, {'id': 'S50', 'elements': ['H', 'K'], 'cost': 92}, {'id': 'S51', 'elements': ['L', 'M', 'P', 'Q'], 'cost': 260}, {'id': 'S52', 'elements': ['C', 'E', 'F', 'G', 'H'], 'cost': 90}, {'id': 'S53', 'elements': ['F', 'H', 'I', 'J', 'K'], 'cost': 350}, {'id': 'S54', 'elements': ['E', 'F', 'G', 'H'], 'cost': 168}, {'id': 'S55', 'elements': ['L', 'N', 'O', 'Q'], 'cost': 80}, {'id': 'S56', 'elements': ['G', 'I', 'J', 'K', 'L'], 'cost': 125}, {'id': 'S57', 'elements': ['A', 'B', 'C', 'E', 'F'], 'cost': 10}, {'id': 'S58', 'elements': ['H', 'I', 'J', 'L'], 'cost': 248}, {'id': 'S59', 'elements': ['A', 'B', 'C', 'F'], 'cost': 132}, {'id': 'S60', 'elements': ['L', 'M', 'N', 'O'], 'cost': 180}, {'id': 'S61', 'elements': ['L', 'N', 'O', 'P', 'Q'], 'cost': 165}, {'id': 'S62', 'elements': ['A', 'B', 'D', 'E'], 'cost': 196}, {'id': 'S63', 'elements': ['F', 'G', 'H', 'I'], 'cost': 364}, {'id': 'S64', 'elements': ['P', 'Q'], 'cost': 120}, {'id': 'S65', 'elements': ['M', 'N', 'O'], 'cost': 99}, {'id': 'S66', 'elements': ['K', 'L', 'O'], 'cost': 30}, {'id': 'S67', 'elements': ['O', 'P'], 'cost': 192}, {'id': 'S68', 'elements': ['C', 'E', 'F', 'G'], 'cost': 352}, {'id': 'S69', 'elements': ['C', 'H'], 'cost': 156}, {'id': 'S70', 'elements': ['J', 'K', 'L', 'M', 'N'], 'cost': 275}, {'id': 'S71', 'elements': ['A'], 'cost': 10000}, {'id': 'S72', 'elements': ['B'], 'cost': 10000}, {'id': 'S73', 'elements': ['C'], 'cost': 10000}, {'id': 'S74', 'elements': ['D'], 'cost': 10000}, {'id': 'S75', 'elements': ['E'], 'cost': 10000}, {'id': 'S76', 'elements': ['F'], 'cost': 10000}, {'id': 'S77', 'elements': ['G'], 'cost': 10000}, {'id': 'S78', 'elements': ['H'], 'cost': 10000}, {'id': 'S79', 'elements': ['I'], 'cost': 10000}, {'id': 'S80', 'elements': ['J'], 'cost': 10000}, {'id': 'S81', 'elements': ['K'], 'cost': 10000}, {'id': 'S82', 'elements': ['L'], 'cost': 10000}, {'id': 'S83', 'elements': ['M'], 'cost': 10000}, {'id': 'S84', 'elements': ['N'], 'cost': 10000}, {'id': 'S85', 'elements': ['O'], 'cost': 10000}, {'id': 'S86', 'elements': ['P'], 'cost': 10000}, {'id': 'S87', 'elements': ['Q'], 'cost': 10000}]}","['S1', 'S11', 'S15', 'S18', 'S56']",41,markdown_table,names
SPP,SPP,"There’s a local relay coming up and the organizer’s juggling which squads to hire: each squad covers certain parts of the course and charges a stipend. The aim is to assign squads so every section is covered by exactly one squad, squads don’t overlap on any section, and the overall payout — add together each hired squad’s stipend — is minimized. The concrete squads, segments, and amounts are given below.

# num_course_segments=13
# num_candidate_squads=43
squad_id,stipend,covered_segments
S1,64,10 11
S2,232,2 3 4 5
S3,255,7 8 9
S4,52,5 6
S5,96,2 4
S6,68,10
S7,123,9 10 12
S8,52,6 7 8 9
S9,20,9 10
S10,96,8 9 11
S11,102,6 7 8
S12,200,12 13
S13,34,9 11
S14,246,11 12 13
S15,9,8
S16,144,10 12 13
S17,244,4 5 6 7
S18,183,10 11 12
S19,134,11 12
S20,111,1 2 3
S21,219,3 5 6
S22,174,4 5
S23,198,9 12
S24,126,3 4 5
S25,186,7 8
S26,132,4 5 7
S27,102,10 12
S28,142,1 5
S29,284,1 2 3 4
S30,165,4 5 6
S31,10000,1
S32,10000,2
S33,10000,3
S34,10000,4
S35,10000,5
S36,10000,6
S37,10000,7
S38,10000,8
S39,10000,9
S40,10000,10
S41,10000,11
S42,10000,12
S43,10000,13

Also, when you show which squads you picked, please use this simple JSON layout so it's easy to read:

{
  ""solution"": [""squad_id"", ...]
}

Here ""solution"" is just a list of the squad IDs you want to hire (one ID per entry). This JSON is only a sketch of the shape I expect — not the final answer itself — so fill that array with the actual squad identifiers from the instance.

Please use the identifiers exactly as they appear in the instance input — no renaming and no new labels. 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 13, 'num_sets': 43, 'density': 0.16279069767441862, 'sets': [{'id': 1, 'elements': [10, 11], 'cost': 64}, {'id': 2, 'elements': [2, 3, 4, 5], 'cost': 232}, {'id': 3, 'elements': [7, 8, 9], 'cost': 255}, {'id': 4, 'elements': [5, 6], 'cost': 52}, {'id': 5, 'elements': [2, 4], 'cost': 96}, {'id': 6, 'elements': [10], 'cost': 68}, {'id': 7, 'elements': [9, 10, 12], 'cost': 123}, {'id': 8, 'elements': [6, 7, 8, 9], 'cost': 52}, {'id': 9, 'elements': [9, 10], 'cost': 20}, {'id': 10, 'elements': [8, 9, 11], 'cost': 96}, {'id': 11, 'elements': [6, 7, 8], 'cost': 102}, {'id': 12, 'elements': [12, 13], 'cost': 200}, {'id': 13, 'elements': [9, 11], 'cost': 34}, {'id': 14, 'elements': [11, 12, 13], 'cost': 246}, {'id': 15, 'elements': [8], 'cost': 9}, {'id': 16, 'elements': [10, 12, 13], 'cost': 144}, {'id': 17, 'elements': [4, 5, 6, 7], 'cost': 244}, {'id': 18, 'elements': [10, 11, 12], 'cost': 183}, {'id': 19, 'elements': [11, 12], 'cost': 134}, {'id': 20, 'elements': [1, 2, 3], 'cost': 111}, {'id': 21, 'elements': [3, 5, 6], 'cost': 219}, {'id': 22, 'elements': [4, 5], 'cost': 174}, {'id': 23, 'elements': [9, 12], 'cost': 198}, {'id': 24, 'elements': [3, 4, 5], 'cost': 126}, {'id': 25, 'elements': [7, 8], 'cost': 186}, {'id': 26, 'elements': [4, 5, 7], 'cost': 132}, {'id': 27, 'elements': [10, 12], 'cost': 102}, {'id': 28, 'elements': [1, 5], 'cost': 142}, {'id': 29, 'elements': [1, 2, 3, 4], 'cost': 284}, {'id': 30, 'elements': [4, 5, 6], 'cost': 165}, {'id': 31, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 32, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 33, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 34, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 35, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 36, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 37, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 38, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 39, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 40, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 41, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 42, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 43, 'elements': [13], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0041_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0041_bag_stats.png'}","[13, 15, 16, 17, 20]",542.0,"{'num_elements': 13, 'num_sets': 43, 'sets': [{'id': 'S1', 'elements': [10, 11], 'cost': 64}, {'id': 'S2', 'elements': [2, 3, 4, 5], 'cost': 232}, {'id': 'S3', 'elements': [7, 8, 9], 'cost': 255}, {'id': 'S4', 'elements': [5, 6], 'cost': 52}, {'id': 'S5', 'elements': [2, 4], 'cost': 96}, {'id': 'S6', 'elements': [10], 'cost': 68}, {'id': 'S7', 'elements': [9, 10, 12], 'cost': 123}, {'id': 'S8', 'elements': [6, 7, 8, 9], 'cost': 52}, {'id': 'S9', 'elements': [9, 10], 'cost': 20}, {'id': 'S10', 'elements': [8, 9, 11], 'cost': 96}, {'id': 'S11', 'elements': [6, 7, 8], 'cost': 102}, {'id': 'S12', 'elements': [12, 13], 'cost': 200}, {'id': 'S13', 'elements': [9, 11], 'cost': 34}, {'id': 'S14', 'elements': [11, 12, 13], 'cost': 246}, {'id': 'S15', 'elements': [8], 'cost': 9}, {'id': 'S16', 'elements': [10, 12, 13], 'cost': 144}, {'id': 'S17', 'elements': [4, 5, 6, 7], 'cost': 244}, {'id': 'S18', 'elements': [10, 11, 12], 'cost': 183}, {'id': 'S19', 'elements': [11, 12], 'cost': 134}, {'id': 'S20', 'elements': [1, 2, 3], 'cost': 111}, {'id': 'S21', 'elements': [3, 5, 6], 'cost': 219}, {'id': 'S22', 'elements': [4, 5], 'cost': 174}, {'id': 'S23', 'elements': [9, 12], 'cost': 198}, {'id': 'S24', 'elements': [3, 4, 5], 'cost': 126}, {'id': 'S25', 'elements': [7, 8], 'cost': 186}, {'id': 'S26', 'elements': [4, 5, 7], 'cost': 132}, {'id': 'S27', 'elements': [10, 12], 'cost': 102}, {'id': 'S28', 'elements': [1, 5], 'cost': 142}, {'id': 'S29', 'elements': [1, 2, 3, 4], 'cost': 284}, {'id': 'S30', 'elements': [4, 5, 6], 'cost': 165}, {'id': 'S31', 'elements': [1], 'cost': 10000}, {'id': 'S32', 'elements': [2], 'cost': 10000}, {'id': 'S33', 'elements': [3], 'cost': 10000}, {'id': 'S34', 'elements': [4], 'cost': 10000}, {'id': 'S35', 'elements': [5], 'cost': 10000}, {'id': 'S36', 'elements': [6], 'cost': 10000}, {'id': 'S37', 'elements': [7], 'cost': 10000}, {'id': 'S38', 'elements': [8], 'cost': 10000}, {'id': 'S39', 'elements': [9], 'cost': 10000}, {'id': 'S40', 'elements': [10], 'cost': 10000}, {'id': 'S41', 'elements': [11], 'cost': 10000}, {'id': 'S42', 'elements': [12], 'cost': 10000}, {'id': 'S43', 'elements': [13], 'cost': 10000}]}","['S13', 'S15', 'S16', 'S17', 'S20']",42,csv,1
SPP,SPP,"A friend is tossing around a few outfit kits before a vacation; each kit covers certain days and contributes a bit of weight to the luggage. The goal is to pick a set of kits so each day is assigned to one—and only one—of the chosen kits, and no chosen kits clash on the same day. The practical criterion is the lightest possible bag: just add up the weights of the chosen kits and aim for the lowest total. The exact options and schedule appear below.

- **total_travel_days**: 17
- **available_kits_count**: 65

| kit_identifier | kit_weight | kit_covered_days |
|---|---|---|
| S1 | 58 | 7 8 |
| S2 | 300 | 4 6 7 10 |
| S3 | 210 | 5 7 10 |
| S4 | 252 | 3 4 6 9 |
| S5 | 118 | 6 9 |
| S6 | 224 | 0 4 6 7 |
| S7 | 228 | 10 11 12 14 |
| S8 | 294 | 8 13 14 |
| S9 | 9 | 13 14 16 |
| S10 | 273 | 11 12 14 |
| S11 | 56 | 10 12 |
| S12 | 147 | 11 13 15 |
| S13 | 36 | 13 14 15 16 |
| S14 | 189 | 0 1 2 |
| S15 | 90 | 12 13 16 |
| S16 | 249 | 6 8 10 |
| S17 | 141 | 14 15 16 |
| S18 | 24 | 5 9 |
| S19 | 76 | 0 1 2 3 |
| S20 | 60 | 5 6 7 8 10 |
| S21 | 12 | 2 4 5 6 |
| S22 | 92 | 3 5 6 8 |
| S23 | 96 | 12 13 14 15 |
| S24 | 96 | 3 5 |
| S25 | 84 | 7 8 9 10 |
| S26 | 332 | 11 12 13 15 |
| S27 | 265 | 9 10 11 12 13 |
| S28 | 20 | 12 13 |
| S29 | 156 | 9 10 12 13 |
| S30 | 300 | 8 10 11 |
| S31 | 156 | 1 4 5 6 |
| S32 | 189 | 1 3 4 |
| S33 | 21 | 12 13 14 |
| S34 | 21 | 3 5 6 |
| S35 | 114 | 9 13 14 |
| S36 | 186 | 12 15 16 |
| S37 | 44 | 9 13 |
| S38 | 70 | 1 4 |
| S39 | 8 | 14 15 |
| S40 | 111 | 9 10 11 |
| S41 | 219 | 13 15 16 |
| S42 | 171 | 3 5 7 |
| S43 | 144 | 6 7 8 9 |
| S44 | 213 | 4 6 8 |
| S45 | 248 | 1 4 5 8 |
| S46 | 44 | 6 7 9 11 |
| S47 | 136 | 10 12 13 14 |
| S48 | 368 | 8 9 11 12 |
| S49 | 10000 | 0 |
| S50 | 10000 | 1 |
| S51 | 10000 | 2 |
| S52 | 10000 | 3 |
| S53 | 10000 | 4 |
| S54 | 10000 | 5 |
| S55 | 10000 | 6 |
| S56 | 10000 | 7 |
| S57 | 10000 | 8 |
| S58 | 10000 | 9 |
| S59 | 10000 | 10 |
| S60 | 10000 | 11 |
| S61 | 10000 | 12 |
| S62 | 10000 | 13 |
| S63 | 10000 | 14 |
| S64 | 10000 | 15 |
| S65 | 10000 | 16 |

If you want to hand me the chosen kits in a simple, predictable way, just use this little JSON shape when you reply:

{
  ""solution"": [""kit_id"", ...]
}

""solution"" is the list of the outfit kit identifiers you picked — one entry per chosen kit. Think of it like filling in a short form: drop in the exact kit IDs from the instance so I can read them straight away. This is just a sketch of the shape I expect, not the actual answer.

Please make sure every identifier is used exactly as it appears in the instance input — no renaming and no new labels. 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 17, 'num_sets': 65, 'density': 0.15746606334841629, 'sets': [{'id': 1, 'elements': [8, 9], 'cost': 58}, {'id': 2, 'elements': [5, 7, 8, 11], 'cost': 300}, {'id': 3, 'elements': [6, 8, 11], 'cost': 210}, {'id': 4, 'elements': [4, 5, 7, 10], 'cost': 252}, {'id': 5, 'elements': [7, 10], 'cost': 118}, {'id': 6, 'elements': [1, 5, 7, 8], 'cost': 224}, {'id': 7, 'elements': [11, 12, 13, 15], 'cost': 228}, {'id': 8, 'elements': [9, 14, 15], 'cost': 294}, {'id': 9, 'elements': [14, 15, 17], 'cost': 9}, {'id': 10, 'elements': [12, 13, 15], 'cost': 273}, {'id': 11, 'elements': [11, 13], 'cost': 56}, {'id': 12, 'elements': [12, 14, 16], 'cost': 147}, {'id': 13, 'elements': [14, 15, 16, 17], 'cost': 36}, {'id': 14, 'elements': [1, 2, 3], 'cost': 189}, {'id': 15, 'elements': [13, 14, 17], 'cost': 90}, {'id': 16, 'elements': [7, 9, 11], 'cost': 249}, {'id': 17, 'elements': [15, 16, 17], 'cost': 141}, {'id': 18, 'elements': [6, 10], 'cost': 24}, {'id': 19, 'elements': [1, 2, 3, 4], 'cost': 76}, {'id': 20, 'elements': [6, 7, 8, 9, 11], 'cost': 60}, {'id': 21, 'elements': [3, 5, 6, 7], 'cost': 12}, {'id': 22, 'elements': [4, 6, 7, 9], 'cost': 92}, {'id': 23, 'elements': [13, 14, 15, 16], 'cost': 96}, {'id': 24, 'elements': [4, 6], 'cost': 96}, {'id': 25, 'elements': [8, 9, 10, 11], 'cost': 84}, {'id': 26, 'elements': [12, 13, 14, 16], 'cost': 332}, {'id': 27, 'elements': [10, 11, 12, 13, 14], 'cost': 265}, {'id': 28, 'elements': [13, 14], 'cost': 20}, {'id': 29, 'elements': [10, 11, 13, 14], 'cost': 156}, {'id': 30, 'elements': [9, 11, 12], 'cost': 300}, {'id': 31, 'elements': [2, 5, 6, 7], 'cost': 156}, {'id': 32, 'elements': [2, 4, 5], 'cost': 189}, {'id': 33, 'elements': [13, 14, 15], 'cost': 21}, {'id': 34, 'elements': [4, 6, 7], 'cost': 21}, {'id': 35, 'elements': [10, 14, 15], 'cost': 114}, {'id': 36, 'elements': [13, 16, 17], 'cost': 186}, {'id': 37, 'elements': [10, 14], 'cost': 44}, {'id': 38, 'elements': [2, 5], 'cost': 70}, {'id': 39, 'elements': [15, 16], 'cost': 8}, {'id': 40, 'elements': [10, 11, 12], 'cost': 111}, {'id': 41, 'elements': [14, 16, 17], 'cost': 219}, {'id': 42, 'elements': [4, 6, 8], 'cost': 171}, {'id': 43, 'elements': [7, 8, 9, 10], 'cost': 144}, {'id': 44, 'elements': [5, 7, 9], 'cost': 213}, {'id': 45, 'elements': [2, 5, 6, 9], 'cost': 248}, {'id': 46, 'elements': [7, 8, 10, 12], 'cost': 44}, {'id': 47, 'elements': [11, 13, 14, 15], 'cost': 136}, {'id': 48, 'elements': [9, 10, 12, 13], 'cost': 368}, {'id': 49, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [17], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0042_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0042_bag_stats.png'}","[14, 15, 39, 40, 42, 44]",782.0,"{'num_elements': 17, 'num_sets': 65, 'sets': [{'id': 'S1', 'elements': [7, 8], 'cost': 58}, {'id': 'S2', 'elements': [4, 6, 7, 10], 'cost': 300}, {'id': 'S3', 'elements': [5, 7, 10], 'cost': 210}, {'id': 'S4', 'elements': [3, 4, 6, 9], 'cost': 252}, {'id': 'S5', 'elements': [6, 9], 'cost': 118}, {'id': 'S6', 'elements': [0, 4, 6, 7], 'cost': 224}, {'id': 'S7', 'elements': [10, 11, 12, 14], 'cost': 228}, {'id': 'S8', 'elements': [8, 13, 14], 'cost': 294}, {'id': 'S9', 'elements': [13, 14, 16], 'cost': 9}, {'id': 'S10', 'elements': [11, 12, 14], 'cost': 273}, {'id': 'S11', 'elements': [10, 12], 'cost': 56}, {'id': 'S12', 'elements': [11, 13, 15], 'cost': 147}, {'id': 'S13', 'elements': [13, 14, 15, 16], 'cost': 36}, {'id': 'S14', 'elements': [0, 1, 2], 'cost': 189}, {'id': 'S15', 'elements': [12, 13, 16], 'cost': 90}, {'id': 'S16', 'elements': [6, 8, 10], 'cost': 249}, {'id': 'S17', 'elements': [14, 15, 16], 'cost': 141}, {'id': 'S18', 'elements': [5, 9], 'cost': 24}, {'id': 'S19', 'elements': [0, 1, 2, 3], 'cost': 76}, {'id': 'S20', 'elements': [5, 6, 7, 8, 10], 'cost': 60}, {'id': 'S21', 'elements': [2, 4, 5, 6], 'cost': 12}, {'id': 'S22', 'elements': [3, 5, 6, 8], 'cost': 92}, {'id': 'S23', 'elements': [12, 13, 14, 15], 'cost': 96}, {'id': 'S24', 'elements': [3, 5], 'cost': 96}, {'id': 'S25', 'elements': [7, 8, 9, 10], 'cost': 84}, {'id': 'S26', 'elements': [11, 12, 13, 15], 'cost': 332}, {'id': 'S27', 'elements': [9, 10, 11, 12, 13], 'cost': 265}, {'id': 'S28', 'elements': [12, 13], 'cost': 20}, {'id': 'S29', 'elements': [9, 10, 12, 13], 'cost': 156}, {'id': 'S30', 'elements': [8, 10, 11], 'cost': 300}, {'id': 'S31', 'elements': [1, 4, 5, 6], 'cost': 156}, {'id': 'S32', 'elements': [1, 3, 4], 'cost': 189}, {'id': 'S33', 'elements': [12, 13, 14], 'cost': 21}, {'id': 'S34', 'elements': [3, 5, 6], 'cost': 21}, {'id': 'S35', 'elements': [9, 13, 14], 'cost': 114}, {'id': 'S36', 'elements': [12, 15, 16], 'cost': 186}, {'id': 'S37', 'elements': [9, 13], 'cost': 44}, {'id': 'S38', 'elements': [1, 4], 'cost': 70}, {'id': 'S39', 'elements': [14, 15], 'cost': 8}, {'id': 'S40', 'elements': [9, 10, 11], 'cost': 111}, {'id': 'S41', 'elements': [13, 15, 16], 'cost': 219}, {'id': 'S42', 'elements': [3, 5, 7], 'cost': 171}, {'id': 'S43', 'elements': [6, 7, 8, 9], 'cost': 144}, {'id': 'S44', 'elements': [4, 6, 8], 'cost': 213}, {'id': 'S45', 'elements': [1, 4, 5, 8], 'cost': 248}, {'id': 'S46', 'elements': [6, 7, 9, 11], 'cost': 44}, {'id': 'S47', 'elements': [10, 12, 13, 14], 'cost': 136}, {'id': 'S48', 'elements': [8, 9, 11, 12], 'cost': 368}, {'id': 'S49', 'elements': [0], 'cost': 10000}, {'id': 'S50', 'elements': [1], 'cost': 10000}, {'id': 'S51', 'elements': [2], 'cost': 10000}, {'id': 'S52', 'elements': [3], 'cost': 10000}, {'id': 'S53', 'elements': [4], 'cost': 10000}, {'id': 'S54', 'elements': [5], 'cost': 10000}, {'id': 'S55', 'elements': [6], 'cost': 10000}, {'id': 'S56', 'elements': [7], 'cost': 10000}, {'id': 'S57', 'elements': [8], 'cost': 10000}, {'id': 'S58', 'elements': [9], 'cost': 10000}, {'id': 'S59', 'elements': [10], 'cost': 10000}, {'id': 'S60', 'elements': [11], 'cost': 10000}, {'id': 'S61', 'elements': [12], 'cost': 10000}, {'id': 'S62', 'elements': [13], 'cost': 10000}, {'id': 'S63', 'elements': [14], 'cost': 10000}, {'id': 'S64', 'elements': [15], 'cost': 10000}, {'id': 'S65', 'elements': [16], 'cost': 10000}]}","['S14', 'S15', 'S39', 'S40', 'S42', 'S44']",43,markdown_table,0
SPP,SPP,"I’m running the relief side of things and need to decide which pre-packed medical boxes to send out so that every kind of treatment needed in the field is included once and only once — no two boxes should cover the same treatment, and nothing can be left out. The aim is to keep the total shipping bill as small as possible, which is just the sum of the shipping fees for whatever boxes get sent. Concrete details about the available packs and their costs are listed below.

I have 17 treatment categories to cover exactly once and 75 pre-packed medical boxes available; details follow:
Pack S1: covers treatments 4 7 8; shipping fee 129.
Pack S2: covers treatments 8 9 10 11 14; shipping fee 165.
Pack S3: covers treatments 5 7; shipping fee 58.
Pack S4: covers treatments 5 6 7 8 9; shipping fee 480.
Pack S5: covers treatments 10 11 12; shipping fee 255.
Pack S6: covers treatments 8 13; shipping fee 104.
Pack S7: covers treatments 4 5 6 9; shipping fee 160.
Pack S8: covers treatments 14 15 17; shipping fee 168.
Pack S9: covers treatments 8 9 10 11 12; shipping fee 55.
Pack S10: covers treatments 1 4; shipping fee 140.
Pack S11: covers treatments 3 4 6; shipping fee 60.
Pack S12: covers treatments 1 2 3 4 5; shipping fee 450.
Pack S13: covers treatments 1 3 5; shipping fee 15.
Pack S14: covers treatments 1 2 4 5 7; shipping fee 255.
Pack S15: covers treatments 13 14 15 16 17; shipping fee 185.
Pack S16: covers treatments 13 15; shipping fee 166.
Pack S17: covers treatments 13 14 16 17; shipping fee 176.
Pack S18: covers treatments 13 15 16 17; shipping fee 364.
Pack S19: covers treatments 12 13 15 17; shipping fee 168.
Pack S20: covers treatments 13 14; shipping fee 22.
Pack S21: covers treatments 2 9; shipping fee 34.
Pack S22: covers treatments 2 3; shipping fee 82.
Pack S23: covers treatments 14 15 16 17; shipping fee 284.
Pack S24: covers treatments 6 9 10 13; shipping fee 84.
Pack S25: covers treatments 9 10 14; shipping fee 99.
Pack S26: covers treatments 15 16 17; shipping fee 126.
Pack S27: covers treatments 2 5 6 7 9; shipping fee 215.
Pack S28: covers treatments 4 6; shipping fee 72.
Pack S29: covers treatments 6 7 8 9 10; shipping fee 355.
Pack S30: covers treatments 12 14 15; shipping fee 243.
Pack S31: covers treatments 2 3 4 5 6; shipping fee 150.
Pack S32: covers treatments 4 6 7 8 9; shipping fee 375.
Pack S33: covers treatments 5 8 9; shipping fee 174.
Pack S34: covers treatments 2 3 5; shipping fee 108.
Pack S35: covers treatments 10 14 15 16; shipping fee 160.
Pack S36: covers treatments 1 2 3 4; shipping fee 400.
Pack S37: covers treatments 7 10; shipping fee 186.
Pack S38: covers treatments 3 6 7; shipping fee 96.
Pack S39: covers treatments 4 5 8; shipping fee 39.
Pack S40: covers treatments 7 9 10 11; shipping fee 80.
Pack S41: covers treatments 14 15; shipping fee 200.
Pack S42: covers treatments 10 11 12 14; shipping fee 12.
Pack S43: covers treatments 2 3 4 5 7; shipping fee 125.
Pack S44: covers treatments 9 13 14; shipping fee 168.
Pack S45: covers treatments 15 17; shipping fee 44.
Pack S46: covers treatments 16 17; shipping fee 18.
Pack S47: covers treatments 7 8 9 10 11; shipping fee 210.
Pack S48: covers treatments 4 6 7 9; shipping fee 12.
Pack S49: covers treatments 11 12 13 14 16; shipping fee 55.
Pack S50: covers treatments 4 5; shipping fee 30.
Pack S51: covers treatments 1 2 4; shipping fee 54.
Pack S52: covers treatments 12 15 16 17; shipping fee 272.
Pack S53: covers treatments 9 10 11 12 13; shipping fee 65.
Pack S54: covers treatments 3 5 7; shipping fee 48.
Pack S55: covers treatments 4 7; shipping fee 14.
Pack S56: covers treatments 3 4 5; shipping fee 267.
Pack S57: covers treatments 8 9 10 13; shipping fee 216.
Pack S58: covers treatments 8 10; shipping fee 18.
Pack S59: covers treatments 1; shipping fee 10000.
Pack S60: covers treatments 2; shipping fee 10000.
Pack S61: covers treatments 3; shipping fee 10000.
Pack S62: covers treatments 4; shipping fee 10000.
Pack S63: covers treatments 5; shipping fee 10000.
Pack S64: covers treatments 6; shipping fee 10000.
Pack S65: covers treatments 7; shipping fee 10000.
Pack S66: covers treatments 8; shipping fee 10000.
Pack S67: covers treatments 9; shipping fee 10000.
Pack S68: covers treatments 10; shipping fee 10000.
Pack S69: covers treatments 11; shipping fee 10000.
Pack S70: covers treatments 12; shipping fee 10000.
Pack S71: covers treatments 13; shipping fee 10000.
Pack S72: covers treatments 14; shipping fee 10000.
Pack S73: covers treatments 15; shipping fee 10000.
Pack S74: covers treatments 16; shipping fee 10000.
Pack S75: covers treatments 17; shipping fee 10000.
I'll pick packs so each of the 17 treatments is covered exactly once while minimizing total shipping fees.

Also, when you send your pick, please use this simple JSON layout so it's easy to read and process — just drop the chosen box IDs in the array like this:

{
  ""solution"": [""box_id"", ...]
}

This just shows the shape I expect: ""solution"" is the list of pre-packed medical boxes you want to send. The ""box_id"" placeholder stands in for whatever actual box IDs you pick from the instance; replace it with the real IDs when you answer. The JSON above is only a sketch of the expected format, not the final answer.

All identifiers must be used exactly as they appear in the instance input — no renaming and no new labels.
Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'problem_type': 'SPP', 'num_elements': 17, 'num_sets': 75, 'density': 0.17019607843137255, 'sets': [{'id': 1, 'elements': [4, 7, 8], 'cost': 129}, {'id': 2, 'elements': [8, 9, 10, 11, 14], 'cost': 165}, {'id': 3, 'elements': [5, 7], 'cost': 58}, {'id': 4, 'elements': [5, 6, 7, 8, 9], 'cost': 480}, {'id': 5, 'elements': [10, 11, 12], 'cost': 255}, {'id': 6, 'elements': [8, 13], 'cost': 104}, {'id': 7, 'elements': [4, 5, 6, 9], 'cost': 160}, {'id': 8, 'elements': [14, 15, 17], 'cost': 168}, {'id': 9, 'elements': [8, 9, 10, 11, 12], 'cost': 55}, {'id': 10, 'elements': [1, 4], 'cost': 140}, {'id': 11, 'elements': [3, 4, 6], 'cost': 60}, {'id': 12, 'elements': [1, 2, 3, 4, 5], 'cost': 450}, {'id': 13, 'elements': [1, 3, 5], 'cost': 15}, {'id': 14, 'elements': [1, 2, 4, 5, 7], 'cost': 255}, {'id': 15, 'elements': [13, 14, 15, 16, 17], 'cost': 185}, {'id': 16, 'elements': [13, 15], 'cost': 166}, {'id': 17, 'elements': [13, 14, 16, 17], 'cost': 176}, {'id': 18, 'elements': [13, 15, 16, 17], 'cost': 364}, {'id': 19, 'elements': [12, 13, 15, 17], 'cost': 168}, {'id': 20, 'elements': [13, 14], 'cost': 22}, {'id': 21, 'elements': [2, 9], 'cost': 34}, {'id': 22, 'elements': [2, 3], 'cost': 82}, {'id': 23, 'elements': [14, 15, 16, 17], 'cost': 284}, {'id': 24, 'elements': [6, 9, 10, 13], 'cost': 84}, {'id': 25, 'elements': [9, 10, 14], 'cost': 99}, {'id': 26, 'elements': [15, 16, 17], 'cost': 126}, {'id': 27, 'elements': [2, 5, 6, 7, 9], 'cost': 215}, {'id': 28, 'elements': [4, 6], 'cost': 72}, {'id': 29, 'elements': [6, 7, 8, 9, 10], 'cost': 355}, {'id': 30, 'elements': [12, 14, 15], 'cost': 243}, {'id': 31, 'elements': [2, 3, 4, 5, 6], 'cost': 150}, {'id': 32, 'elements': [4, 6, 7, 8, 9], 'cost': 375}, {'id': 33, 'elements': [5, 8, 9], 'cost': 174}, {'id': 34, 'elements': [2, 3, 5], 'cost': 108}, {'id': 35, 'elements': [10, 14, 15, 16], 'cost': 160}, {'id': 36, 'elements': [1, 2, 3, 4], 'cost': 400}, {'id': 37, 'elements': [7, 10], 'cost': 186}, {'id': 38, 'elements': [3, 6, 7], 'cost': 96}, {'id': 39, 'elements': [4, 5, 8], 'cost': 39}, {'id': 40, 'elements': [7, 9, 10, 11], 'cost': 80}, {'id': 41, 'elements': [14, 15], 'cost': 200}, {'id': 42, 'elements': [10, 11, 12, 14], 'cost': 12}, {'id': 43, 'elements': [2, 3, 4, 5, 7], 'cost': 125}, {'id': 44, 'elements': [9, 13, 14], 'cost': 168}, {'id': 45, 'elements': [15, 17], 'cost': 44}, {'id': 46, 'elements': [16, 17], 'cost': 18}, {'id': 47, 'elements': [7, 8, 9, 10, 11], 'cost': 210}, {'id': 48, 'elements': [4, 6, 7, 9], 'cost': 12}, {'id': 49, 'elements': [11, 12, 13, 14, 16], 'cost': 55}, {'id': 50, 'elements': [4, 5], 'cost': 30}, {'id': 51, 'elements': [1, 2, 4], 'cost': 54}, {'id': 52, 'elements': [12, 15, 16, 17], 'cost': 272}, {'id': 53, 'elements': [9, 10, 11, 12, 13], 'cost': 65}, {'id': 54, 'elements': [3, 5, 7], 'cost': 48}, {'id': 55, 'elements': [4, 7], 'cost': 14}, {'id': 56, 'elements': [3, 4, 5], 'cost': 267}, {'id': 57, 'elements': [8, 9, 10, 13], 'cost': 216}, {'id': 58, 'elements': [8, 10], 'cost': 18}, {'id': 59, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 66, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 67, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 68, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 69, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 70, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 71, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 72, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 73, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 74, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 75, 'elements': [17], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0043_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0043_bag_stats.png'}","[16, 33, 38, 42, 46, 51]",520.0,"{'num_elements': 17, 'num_sets': 75, 'sets': [{'id': 'S1', 'elements': [4, 7, 8], 'cost': 129}, {'id': 'S2', 'elements': [8, 9, 10, 11, 14], 'cost': 165}, {'id': 'S3', 'elements': [5, 7], 'cost': 58}, {'id': 'S4', 'elements': [5, 6, 7, 8, 9], 'cost': 480}, {'id': 'S5', 'elements': [10, 11, 12], 'cost': 255}, {'id': 'S6', 'elements': [8, 13], 'cost': 104}, {'id': 'S7', 'elements': [4, 5, 6, 9], 'cost': 160}, {'id': 'S8', 'elements': [14, 15, 17], 'cost': 168}, {'id': 'S9', 'elements': [8, 9, 10, 11, 12], 'cost': 55}, {'id': 'S10', 'elements': [1, 4], 'cost': 140}, {'id': 'S11', 'elements': [3, 4, 6], 'cost': 60}, {'id': 'S12', 'elements': [1, 2, 3, 4, 5], 'cost': 450}, {'id': 'S13', 'elements': [1, 3, 5], 'cost': 15}, {'id': 'S14', 'elements': [1, 2, 4, 5, 7], 'cost': 255}, {'id': 'S15', 'elements': [13, 14, 15, 16, 17], 'cost': 185}, {'id': 'S16', 'elements': [13, 15], 'cost': 166}, {'id': 'S17', 'elements': [13, 14, 16, 17], 'cost': 176}, {'id': 'S18', 'elements': [13, 15, 16, 17], 'cost': 364}, {'id': 'S19', 'elements': [12, 13, 15, 17], 'cost': 168}, {'id': 'S20', 'elements': [13, 14], 'cost': 22}, {'id': 'S21', 'elements': [2, 9], 'cost': 34}, {'id': 'S22', 'elements': [2, 3], 'cost': 82}, {'id': 'S23', 'elements': [14, 15, 16, 17], 'cost': 284}, {'id': 'S24', 'elements': [6, 9, 10, 13], 'cost': 84}, {'id': 'S25', 'elements': [9, 10, 14], 'cost': 99}, {'id': 'S26', 'elements': [15, 16, 17], 'cost': 126}, {'id': 'S27', 'elements': [2, 5, 6, 7, 9], 'cost': 215}, {'id': 'S28', 'elements': [4, 6], 'cost': 72}, {'id': 'S29', 'elements': [6, 7, 8, 9, 10], 'cost': 355}, {'id': 'S30', 'elements': [12, 14, 15], 'cost': 243}, {'id': 'S31', 'elements': [2, 3, 4, 5, 6], 'cost': 150}, {'id': 'S32', 'elements': [4, 6, 7, 8, 9], 'cost': 375}, {'id': 'S33', 'elements': [5, 8, 9], 'cost': 174}, {'id': 'S34', 'elements': [2, 3, 5], 'cost': 108}, {'id': 'S35', 'elements': [10, 14, 15, 16], 'cost': 160}, {'id': 'S36', 'elements': [1, 2, 3, 4], 'cost': 400}, {'id': 'S37', 'elements': [7, 10], 'cost': 186}, {'id': 'S38', 'elements': [3, 6, 7], 'cost': 96}, {'id': 'S39', 'elements': [4, 5, 8], 'cost': 39}, {'id': 'S40', 'elements': [7, 9, 10, 11], 'cost': 80}, {'id': 'S41', 'elements': [14, 15], 'cost': 200}, {'id': 'S42', 'elements': [10, 11, 12, 14], 'cost': 12}, {'id': 'S43', 'elements': [2, 3, 4, 5, 7], 'cost': 125}, {'id': 'S44', 'elements': [9, 13, 14], 'cost': 168}, {'id': 'S45', 'elements': [15, 17], 'cost': 44}, {'id': 'S46', 'elements': [16, 17], 'cost': 18}, {'id': 'S47', 'elements': [7, 8, 9, 10, 11], 'cost': 210}, {'id': 'S48', 'elements': [4, 6, 7, 9], 'cost': 12}, {'id': 'S49', 'elements': [11, 12, 13, 14, 16], 'cost': 55}, {'id': 'S50', 'elements': [4, 5], 'cost': 30}, {'id': 'S51', 'elements': [1, 2, 4], 'cost': 54}, {'id': 'S52', 'elements': [12, 15, 16, 17], 'cost': 272}, {'id': 'S53', 'elements': [9, 10, 11, 12, 13], 'cost': 65}, {'id': 'S54', 'elements': [3, 5, 7], 'cost': 48}, {'id': 'S55', 'elements': [4, 7], 'cost': 14}, {'id': 'S56', 'elements': [3, 4, 5], 'cost': 267}, {'id': 'S57', 'elements': [8, 9, 10, 13], 'cost': 216}, {'id': 'S58', 'elements': [8, 10], 'cost': 18}, {'id': 'S59', 'elements': [1], 'cost': 10000}, {'id': 'S60', 'elements': [2], 'cost': 10000}, {'id': 'S61', 'elements': [3], 'cost': 10000}, {'id': 'S62', 'elements': [4], 'cost': 10000}, {'id': 'S63', 'elements': [5], 'cost': 10000}, {'id': 'S64', 'elements': [6], 'cost': 10000}, {'id': 'S65', 'elements': [7], 'cost': 10000}, {'id': 'S66', 'elements': [8], 'cost': 10000}, {'id': 'S67', 'elements': [9], 'cost': 10000}, {'id': 'S68', 'elements': [10], 'cost': 10000}, {'id': 'S69', 'elements': [11], 'cost': 10000}, {'id': 'S70', 'elements': [12], 'cost': 10000}, {'id': 'S71', 'elements': [13], 'cost': 10000}, {'id': 'S72', 'elements': [14], 'cost': 10000}, {'id': 'S73', 'elements': [15], 'cost': 10000}, {'id': 'S74', 'elements': [16], 'cost': 10000}, {'id': 'S75', 'elements': [17], 'cost': 10000}]}","['S16', 'S33', 'S38', 'S42', 'S46', 'S51']",44,nl,1
SPP,SPP,"There’s a campaign puzzle at hand — pick a set of advertising bundles so that all the target groups are covered, each group gets served by a single chosen bundle, and no two chosen bundles try to reach the same group. The simple measure of success is the total bill: sum the prices of the selected bundles and keep that sum as low as possible. The specific bundles and which segments they touch are listed below.

{
  ""total_customer_segments"": 13,
  ""total_ad_bundles"": 59,
  ""sets"": [
    {
      ""bundle_id"": ""S1"",
      ""target_segments"": [
        ""A"",
        ""C"",
        ""D""
      ],
      ""bundle_price"": 156
    },
    {
      ""bundle_id"": ""S2"",
      ""target_segments"": [
        ""D"",
        ""F"",
        ""G""
      ],
      ""bundle_price"": 96
    },
    {
      ""bundle_id"": ""S3"",
      ""target_segments"": [
        ""B"",
        ""E""
      ],
      ""bundle_price"": 180
    },
    {
      ""bundle_id"": ""S4"",
      ""target_segments"": [
        ""I"",
        ""J"",
        ""K"",
        ""L""
      ],
      ""bundle_price"": 380
    },
    {
      ""bundle_id"": ""S5"",
      ""target_segments"": [
        ""H"",
        ""J"",
        ""K"",
        ""M""
      ],
      ""bundle_price"": 140
    },
    {
      ""bundle_id"": ""S6"",
      ""target_segments"": [
        ""K"",
        ""L"",
        ""M""
      ],
      ""bundle_price"": 24
    },
    {
      ""bundle_id"": ""S7"",
      ""target_segments"": [
        ""I"",
        ""K""
      ],
      ""bundle_price"": 136
    },
    {
      ""bundle_id"": ""S8"",
      ""target_segments"": [
        ""C"",
        ""D""
      ],
      ""bundle_price"": 150
    },
    {
      ""bundle_id"": ""S9"",
      ""target_segments"": [
        ""D"",
        ""F""
      ],
      ""bundle_price"": 182
    },
    {
      ""bundle_id"": ""S10"",
      ""target_segments"": [
        ""G"",
        ""K""
      ],
      ""bundle_price"": 30
    },
    {
      ""bundle_id"": ""S11"",
      ""target_segments"": [
        ""D"",
        ""E""
      ],
      ""bundle_price"": 148
    },
    {
      ""bundle_id"": ""S12"",
      ""target_segments"": [
        ""K"",
        ""M""
      ],
      ""bundle_price"": 64
    },
    {
      ""bundle_id"": ""S13"",
      ""target_segments"": [
        ""A"",
        ""B""
      ],
      ""bundle_price"": 12
    },
    {
      ""bundle_id"": ""S14"",
      ""target_segments"": [
        ""J"",
        ""K"",
        ""M""
      ],
      ""bundle_price"": 45
    },
    {
      ""bundle_id"": ""S15"",
      ""target_segments"": [
        ""I"",
        ""J"",
        ""L""
      ],
      ""bundle_price"": 159
    },
    {
      ""bundle_id"": ""S16"",
      ""target_segments"": [
        ""B"",
        ""D""
      ],
      ""bundle_price"": 166
    },
    {
      ""bundle_id"": ""S17"",
      ""target_segments"": [
        ""E"",
        ""F"",
        ""G""
      ],
      ""bundle_price"": 264
    },
    {
      ""bundle_id"": ""S18"",
      ""target_segments"": [
        ""J"",
        ""L""
      ],
      ""bundle_price"": 54
    },
    {
      ""bundle_id"": ""S19"",
      ""target_segments"": [
        ""E"",
        ""F""
      ],
      ""bundle_price"": 170
    },
    {
      ""bundle_id"": ""S20"",
      ""target_segments"": [
        ""H"",
        ""J""
      ],
      ""bundle_price"": 66
    },
    {
      ""bundle_id"": ""S21"",
      ""target_segments"": [
        ""G""
      ],
      ""bundle_price"": 8
    },
    {
      ""bundle_id"": ""S22"",
      ""target_segments"": [
        ""H"",
        ""I"",
        ""K""
      ],
      ""bundle_price"": 57
    },
    {
      ""bundle_id"": ""S23"",
      ""target_segments"": [
        ""A"",
        ""C""
      ],
      ""bundle_price"": 196
    },
    {
      ""bundle_id"": ""S24"",
      ""target_segments"": [
        ""H"",
        ""I"",
        ""J""
      ],
      ""bundle_price"": 78
    },
    {
      ""bundle_id"": ""S25"",
      ""target_segments"": [
        ""E"",
        ""G"",
        ""J""
      ],
      ""bundle_price"": 189
    },
    {
      ""bundle_id"": ""S26"",
      ""target_segments"": [
        ""K"",
        ""L""
      ],
      ""bundle_price"": 26
    },
    {
      ""bundle_id"": ""S27"",
      ""target_segments"": [
        ""J"",
        ""K""
      ],
      ""bundle_price"": 104
    },
    {
      ""bundle_id"": ""S28"",
      ""target_segments"": [
        ""L""
      ],
      ""bundle_price"": 14
    },
    {
      ""bundle_id"": ""S29"",
      ""target_segments"": [
        ""H"",
        ""I"",
        ""J"",
        ""K""
      ],
      ""bundle_price"": 324
    },
    {
      ""bundle_id"": ""S30"",
      ""target_segments"": [
        ""F"",
        ""H"",
        ""K""
      ],
      ""bundle_price"": 267
    },
    {
      ""bundle_id"": ""S31"",
      ""target_segments"": [
        ""C"",
        ""F"",
        ""G""
      ],
      ""bundle_price"": 42
    },
    {
      ""bundle_id"": ""S32"",
      ""target_segments"": [
        ""J"",
        ""K"",
        ""L""
      ],
      ""bundle_price"": 219
    },
    {
      ""bundle_id"": ""S33"",
      ""target_segments"": [
        ""F"",
        ""G""
      ],
      ""bundle_price"": 64
    },
    {
      ""bundle_id"": ""S34"",
      ""target_segments"": [
        ""A"",
        ""B"",
        ""C"",
        ""E""
      ],
      ""bundle_price"": 116
    },
    {
      ""bundle_id"": ""S35"",
      ""target_segments"": [
        ""A"",
        ""B"",
        ""D""
      ],
      ""bundle_price"": 228
    },
    {
      ""bundle_id"": ""S36"",
      ""target_segments"": [
        ""E"",
        ""G""
      ],
      ""bundle_price"": 54
    },
    {
      ""bundle_id"": ""S37"",
      ""target_segments"": [
        ""A"",
        ""B"",
        ""C""
      ],
      ""bundle_price"": 291
    },
    {
      ""bundle_id"": ""S38"",
      ""target_segments"": [
        ""A"",
        ""C"",
        ""E""
      ],
      ""bundle_price"": 159
    },
    {
      ""bundle_id"": ""S39"",
      ""target_segments"": [
        ""A"",
        ""B"",
        ""C"",
        ""D""
      ],
      ""bundle_price"": 220
    },
    {
      ""bundle_id"": ""S40"",
      ""target_segments"": [
        ""I"",
        ""K"",
        ""L""
      ],
      ""bundle_price"": 63
    },
    {
      ""bundle_id"": ""S41"",
      ""target_segments"": [
        ""H"",
        ""I""
      ],
      ""bundle_price"": 186
    },
    {
      ""bundle_id"": ""S42"",
      ""target_segments"": [
        ""E"",
        ""F"",
        ""I""
      ],
      ""bundle_price"": 183
    },
    {
      ""bundle_id"": ""S43"",
      ""target_segments"": [
        ""E"",
        ""F"",
        ""H""
      ],
      ""bundle_price"": 60
    },
    {
      ""bundle_id"": ""S44"",
      ""target_segments"": [
        ""F"",
        ""H""
      ],
      ""bundle_price"": 70
    },
    {
      ""bundle_id"": ""S45"",
      ""target_segments"": [
        ""H"",
        ""K"",
        ""L"",
        ""M""
      ],
      ""bundle_price"": 276
    },
    {
      ""bundle_id"": ""S46"",
      ""target_segments"": [
        ""D"",
        ""E"",
        ""F""
      ],
      ""bundle_price"": 171
    },
    {
      ""bundle_id"": ""S47"",
      ""target_segments"": [
        ""A""
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S48"",
      ""target_segments"": [
        ""B""
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S49"",
      ""target_segments"": [
        ""C""
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S50"",
      ""target_segments"": [
        ""D""
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S51"",
      ""target_segments"": [
        ""E""
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S52"",
      ""target_segments"": [
        ""F""
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S53"",
      ""target_segments"": [
        ""G""
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S54"",
      ""target_segments"": [
        ""H""
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S55"",
      ""target_segments"": [
        ""I""
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S56"",
      ""target_segments"": [
        ""J""
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S57"",
      ""target_segments"": [
        ""K""
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S58"",
      ""target_segments"": [
        ""L""
      ],
      ""bundle_price"": 10000
    },
    {
      ""bundle_id"": ""S59"",
      ""target_segments"": [
        ""M""
      ],
      ""bundle_price"": 10000
    }
  ]
}

When you send your pick, just drop it into a tiny JSON snippet like this:

{
  ""solution"": [""bundle_id"", ...]
}

""solution"" is just the list of advertising bundles you're choosing — one identifier per chosen bundle. ""bundle_id"" is a placeholder showing the shape: replace each placeholder with an actual bundle identifier from the instance. Think of this as filling out a short form, not a full report.

This is only a sketch of the expected shape, not the actual answer. Please use the exact identifiers from the instance input — no renaming and no new labels.  
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 13, 'num_sets': 59, 'density': 0.17470664928292046, 'sets': [{'id': 1, 'elements': [1, 3, 4], 'cost': 156}, {'id': 2, 'elements': [4, 6, 7], 'cost': 96}, {'id': 3, 'elements': [2, 5], 'cost': 180}, {'id': 4, 'elements': [9, 10, 11, 12], 'cost': 380}, {'id': 5, 'elements': [8, 10, 11, 13], 'cost': 140}, {'id': 6, 'elements': [11, 12, 13], 'cost': 24}, {'id': 7, 'elements': [9, 11], 'cost': 136}, {'id': 8, 'elements': [3, 4], 'cost': 150}, {'id': 9, 'elements': [4, 6], 'cost': 182}, {'id': 10, 'elements': [7, 11], 'cost': 30}, {'id': 11, 'elements': [4, 5], 'cost': 148}, {'id': 12, 'elements': [11, 13], 'cost': 64}, {'id': 13, 'elements': [1, 2], 'cost': 12}, {'id': 14, 'elements': [10, 11, 13], 'cost': 45}, {'id': 15, 'elements': [9, 10, 12], 'cost': 159}, {'id': 16, 'elements': [2, 4], 'cost': 166}, {'id': 17, 'elements': [5, 6, 7], 'cost': 264}, {'id': 18, 'elements': [10, 12], 'cost': 54}, {'id': 19, 'elements': [5, 6], 'cost': 170}, {'id': 20, 'elements': [8, 10], 'cost': 66}, {'id': 21, 'elements': [7], 'cost': 8}, {'id': 22, 'elements': [8, 9, 11], 'cost': 57}, {'id': 23, 'elements': [1, 3], 'cost': 196}, {'id': 24, 'elements': [8, 9, 10], 'cost': 78}, {'id': 25, 'elements': [5, 7, 10], 'cost': 189}, {'id': 26, 'elements': [11, 12], 'cost': 26}, {'id': 27, 'elements': [10, 11], 'cost': 104}, {'id': 28, 'elements': [12], 'cost': 14}, {'id': 29, 'elements': [8, 9, 10, 11], 'cost': 324}, {'id': 30, 'elements': [6, 8, 11], 'cost': 267}, {'id': 31, 'elements': [3, 6, 7], 'cost': 42}, {'id': 32, 'elements': [10, 11, 12], 'cost': 219}, {'id': 33, 'elements': [6, 7], 'cost': 64}, {'id': 34, 'elements': [1, 2, 3, 5], 'cost': 116}, {'id': 35, 'elements': [1, 2, 4], 'cost': 228}, {'id': 36, 'elements': [5, 7], 'cost': 54}, {'id': 37, 'elements': [1, 2, 3], 'cost': 291}, {'id': 38, 'elements': [1, 3, 5], 'cost': 159}, {'id': 39, 'elements': [1, 2, 3, 4], 'cost': 220}, {'id': 40, 'elements': [9, 11, 12], 'cost': 63}, {'id': 41, 'elements': [8, 9], 'cost': 186}, {'id': 42, 'elements': [5, 6, 9], 'cost': 183}, {'id': 43, 'elements': [5, 6, 8], 'cost': 60}, {'id': 44, 'elements': [6, 8], 'cost': 70}, {'id': 45, 'elements': [8, 11, 12, 13], 'cost': 276}, {'id': 46, 'elements': [4, 5, 6], 'cost': 171}, {'id': 47, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [13], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0044_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0044_bag_stats.png'}","[6, 11, 13, 24, 31]",304.0,"{'num_elements': 13, 'num_sets': 59, 'sets': [{'id': 'S1', 'elements': ['A', 'C', 'D'], 'cost': 156}, {'id': 'S2', 'elements': ['D', 'F', 'G'], 'cost': 96}, {'id': 'S3', 'elements': ['B', 'E'], 'cost': 180}, {'id': 'S4', 'elements': ['I', 'J', 'K', 'L'], 'cost': 380}, {'id': 'S5', 'elements': ['H', 'J', 'K', 'M'], 'cost': 140}, {'id': 'S6', 'elements': ['K', 'L', 'M'], 'cost': 24}, {'id': 'S7', 'elements': ['I', 'K'], 'cost': 136}, {'id': 'S8', 'elements': ['C', 'D'], 'cost': 150}, {'id': 'S9', 'elements': ['D', 'F'], 'cost': 182}, {'id': 'S10', 'elements': ['G', 'K'], 'cost': 30}, {'id': 'S11', 'elements': ['D', 'E'], 'cost': 148}, {'id': 'S12', 'elements': ['K', 'M'], 'cost': 64}, {'id': 'S13', 'elements': ['A', 'B'], 'cost': 12}, {'id': 'S14', 'elements': ['J', 'K', 'M'], 'cost': 45}, {'id': 'S15', 'elements': ['I', 'J', 'L'], 'cost': 159}, {'id': 'S16', 'elements': ['B', 'D'], 'cost': 166}, {'id': 'S17', 'elements': ['E', 'F', 'G'], 'cost': 264}, {'id': 'S18', 'elements': ['J', 'L'], 'cost': 54}, {'id': 'S19', 'elements': ['E', 'F'], 'cost': 170}, {'id': 'S20', 'elements': ['H', 'J'], 'cost': 66}, {'id': 'S21', 'elements': ['G'], 'cost': 8}, {'id': 'S22', 'elements': ['H', 'I', 'K'], 'cost': 57}, {'id': 'S23', 'elements': ['A', 'C'], 'cost': 196}, {'id': 'S24', 'elements': ['H', 'I', 'J'], 'cost': 78}, {'id': 'S25', 'elements': ['E', 'G', 'J'], 'cost': 189}, {'id': 'S26', 'elements': ['K', 'L'], 'cost': 26}, {'id': 'S27', 'elements': ['J', 'K'], 'cost': 104}, {'id': 'S28', 'elements': ['L'], 'cost': 14}, {'id': 'S29', 'elements': ['H', 'I', 'J', 'K'], 'cost': 324}, {'id': 'S30', 'elements': ['F', 'H', 'K'], 'cost': 267}, {'id': 'S31', 'elements': ['C', 'F', 'G'], 'cost': 42}, {'id': 'S32', 'elements': ['J', 'K', 'L'], 'cost': 219}, {'id': 'S33', 'elements': ['F', 'G'], 'cost': 64}, {'id': 'S34', 'elements': ['A', 'B', 'C', 'E'], 'cost': 116}, {'id': 'S35', 'elements': ['A', 'B', 'D'], 'cost': 228}, {'id': 'S36', 'elements': ['E', 'G'], 'cost': 54}, {'id': 'S37', 'elements': ['A', 'B', 'C'], 'cost': 291}, {'id': 'S38', 'elements': ['A', 'C', 'E'], 'cost': 159}, {'id': 'S39', 'elements': ['A', 'B', 'C', 'D'], 'cost': 220}, {'id': 'S40', 'elements': ['I', 'K', 'L'], 'cost': 63}, {'id': 'S41', 'elements': ['H', 'I'], 'cost': 186}, {'id': 'S42', 'elements': ['E', 'F', 'I'], 'cost': 183}, {'id': 'S43', 'elements': ['E', 'F', 'H'], 'cost': 60}, {'id': 'S44', 'elements': ['F', 'H'], 'cost': 70}, {'id': 'S45', 'elements': ['H', 'K', 'L', 'M'], 'cost': 276}, {'id': 'S46', 'elements': ['D', 'E', 'F'], 'cost': 171}, {'id': 'S47', 'elements': ['A'], 'cost': 10000}, {'id': 'S48', 'elements': ['B'], 'cost': 10000}, {'id': 'S49', 'elements': ['C'], 'cost': 10000}, {'id': 'S50', 'elements': ['D'], 'cost': 10000}, {'id': 'S51', 'elements': ['E'], 'cost': 10000}, {'id': 'S52', 'elements': ['F'], 'cost': 10000}, {'id': 'S53', 'elements': ['G'], 'cost': 10000}, {'id': 'S54', 'elements': ['H'], 'cost': 10000}, {'id': 'S55', 'elements': ['I'], 'cost': 10000}, {'id': 'S56', 'elements': ['J'], 'cost': 10000}, {'id': 'S57', 'elements': ['K'], 'cost': 10000}, {'id': 'S58', 'elements': ['L'], 'cost': 10000}, {'id': 'S59', 'elements': ['M'], 'cost': 10000}]}","['S6', 'S11', 'S13', 'S24', 'S31']",45,json,names
SPP,SPP,"We’ve got a stack of vendor kits and the office manager needs to decide which kits to order so every department’s supply list is fulfilled by one kit alone. No department can be skipped and no department should receive items from more than one ordered kit — it’s one box per department, no overlaps. The way to judge options is simple: add up the costs of the selected kits and pick the combination with the smallest total bill. The concrete options and their prices are shown below.

{
  ""total_departments"": 16,
  ""total_kits_available"": 73,
  ""sets"": [
    {
      ""kit_id"": ""S1"",
      ""covered_departments"": [
        1,
        2,
        5,
        6
      ],
      ""kit_price"": 276
    },
    {
      ""kit_id"": ""S2"",
      ""covered_departments"": [
        3,
        4,
        5,
        6,
        7
      ],
      ""kit_price"": 35
    },
    {
      ""kit_id"": ""S3"",
      ""covered_departments"": [
        5,
        7,
        9
      ],
      ""kit_price"": 234
    },
    {
      ""kit_id"": ""S4"",
      ""covered_departments"": [
        2,
        3,
        4
      ],
      ""kit_price"": 207
    },
    {
      ""kit_id"": ""S5"",
      ""covered_departments"": [
        7,
        8
      ],
      ""kit_price"": 46
    },
    {
      ""kit_id"": ""S6"",
      ""covered_departments"": [
        14,
        16
      ],
      ""kit_price"": 58
    },
    {
      ""kit_id"": ""S7"",
      ""covered_departments"": [
        1,
        3,
        4,
        5
      ],
      ""kit_price"": 280
    },
    {
      ""kit_id"": ""S8"",
      ""covered_departments"": [
        11,
        12
      ],
      ""kit_price"": 88
    },
    {
      ""kit_id"": ""S9"",
      ""covered_departments"": [
        2,
        3
      ],
      ""kit_price"": 178
    },
    {
      ""kit_id"": ""S10"",
      ""covered_departments"": [
        1,
        2
      ],
      ""kit_price"": 192
    },
    {
      ""kit_id"": ""S11"",
      ""covered_departments"": [
        1,
        2,
        3,
        6
      ],
      ""kit_price"": 276
    },
    {
      ""kit_id"": ""S12"",
      ""covered_departments"": [
        1,
        2,
        3,
        4
      ],
      ""kit_price"": 184
    },
    {
      ""kit_id"": ""S13"",
      ""covered_departments"": [
        2,
        4,
        5
      ],
      ""kit_price"": 114
    },
    {
      ""kit_id"": ""S14"",
      ""covered_departments"": [
        1,
        3
      ],
      ""kit_price"": 172
    },
    {
      ""kit_id"": ""S15"",
      ""covered_departments"": [
        14,
        15,
        16
      ],
      ""kit_price"": 60
    },
    {
      ""kit_id"": ""S16"",
      ""covered_departments"": [
        10,
        12
      ],
      ""kit_price"": 82
    },
    {
      ""kit_id"": ""S17"",
      ""covered_departments"": [
        1,
        2,
        3
      ],
      ""kit_price"": 21
    },
    {
      ""kit_id"": ""S18"",
      ""covered_departments"": [
        8,
        11
      ],
      ""kit_price"": 164
    },
    {
      ""kit_id"": ""S19"",
      ""covered_departments"": [
        10,
        11,
        13
      ],
      ""kit_price"": 147
    },
    {
      ""kit_id"": ""S20"",
      ""covered_departments"": [
        7,
        8,
        10
      ],
      ""kit_price"": 126
    },
    {
      ""kit_id"": ""S21"",
      ""covered_departments"": [
        12,
        13
      ],
      ""kit_price"": 144
    },
    {
      ""kit_id"": ""S22"",
      ""covered_departments"": [
        1,
        2,
        3,
        4,
        5
      ],
      ""kit_price"": 305
    },
    {
      ""kit_id"": ""S23"",
      ""covered_departments"": [
        2,
        3,
        5
      ],
      ""kit_price"": 57
    },
    {
      ""kit_id"": ""S24"",
      ""covered_departments"": [
        9,
        10,
        11,
        12
      ],
      ""kit_price"": 160
    },
    {
      ""kit_id"": ""S25"",
      ""covered_departments"": [
        3,
        7,
        8
      ],
      ""kit_price"": 51
    },
    {
      ""kit_id"": ""S26"",
      ""covered_departments"": [
        13,
        14,
        16
      ],
      ""kit_price"": 33
    },
    {
      ""kit_id"": ""S27"",
      ""covered_departments"": [
        2,
        4,
        6
      ],
      ""kit_price"": 150
    },
    {
      ""kit_id"": ""S28"",
      ""covered_departments"": [
        13,
        14
      ],
      ""kit_price"": 12
    },
    {
      ""kit_id"": ""S29"",
      ""covered_departments"": [
        7,
        9
      ],
      ""kit_price"": 140
    },
    {
      ""kit_id"": ""S30"",
      ""covered_departments"": [
        8,
        9,
        10
      ],
      ""kit_price"": 222
    },
    {
      ""kit_id"": ""S31"",
      ""covered_departments"": [
        8,
        12
      ],
      ""kit_price"": 120
    },
    {
      ""kit_id"": ""S32"",
      ""covered_departments"": [
        8,
        9,
        10,
        11,
        12
      ],
      ""kit_price"": 340
    },
    {
      ""kit_id"": ""S33"",
      ""covered_departments"": [
        9,
        13
      ],
      ""kit_price"": 188
    },
    {
      ""kit_id"": ""S34"",
      ""covered_departments"": [
        3,
        6
      ],
      ""kit_price"": 184
    },
    {
      ""kit_id"": ""S35"",
      ""covered_departments"": [
        13,
        14,
        15,
        16
      ],
      ""kit_price"": 104
    },
    {
      ""kit_id"": ""S36"",
      ""covered_departments"": [
        2,
        4
      ],
      ""kit_price"": 56
    },
    {
      ""kit_id"": ""S37"",
      ""covered_departments"": [
        8,
        16
      ],
      ""kit_price"": 106
    },
    {
      ""kit_id"": ""S38"",
      ""covered_departments"": [
        2,
        3,
        5,
        6
      ],
      ""kit_price"": 392
    },
    {
      ""kit_id"": ""S39"",
      ""covered_departments"": [
        11,
        13,
        14,
        15,
        16
      ],
      ""kit_price"": 340
    },
    {
      ""kit_id"": ""S40"",
      ""covered_departments"": [
        9,
        10,
        12
      ],
      ""kit_price"": 231
    },
    {
      ""kit_id"": ""S41"",
      ""covered_departments"": [
        8,
        9,
        10,
        11
      ],
      ""kit_price"": 176
    },
    {
      ""kit_id"": ""S42"",
      ""covered_departments"": [
        3,
        4,
        6,
        7
      ],
      ""kit_price"": 112
    },
    {
      ""kit_id"": ""S43"",
      ""covered_departments"": [
        3,
        4,
        5,
        6
      ],
      ""kit_price"": 48
    },
    {
      ""kit_id"": ""S44"",
      ""covered_departments"": [
        3,
        4
      ],
      ""kit_price"": 76
    },
    {
      ""kit_id"": ""S45"",
      ""covered_departments"": [
        11,
        16
      ],
      ""kit_price"": 94
    },
    {
      ""kit_id"": ""S46"",
      ""covered_departments"": [
        10,
        11,
        13,
        14
      ],
      ""kit_price"": 384
    },
    {
      ""kit_id"": ""S47"",
      ""covered_departments"": [
        9,
        12,
        13,
        14
      ],
      ""kit_price"": 16
    },
    {
      ""kit_id"": ""S48"",
      ""covered_departments"": [
        4,
        6,
        7,
        8,
        9
      ],
      ""kit_price"": 280
    },
    {
      ""kit_id"": ""S49"",
      ""covered_departments"": [
        4,
        5,
        7,
        9
      ],
      ""kit_price"": 24
    },
    {
      ""kit_id"": ""S50"",
      ""covered_departments"": [
        6,
        12
      ],
      ""kit_price"": 136
    },
    {
      ""kit_id"": ""S51"",
      ""covered_departments"": [
        6,
        7,
        8,
        9,
        10
      ],
      ""kit_price"": 340
    },
    {
      ""kit_id"": ""S52"",
      ""covered_departments"": [
        2,
        4,
        5,
        6
      ],
      ""kit_price"": 312
    },
    {
      ""kit_id"": ""S53"",
      ""covered_departments"": [
        9,
        11,
        12,
        16
      ],
      ""kit_price"": 296
    },
    {
      ""kit_id"": ""S54"",
      ""covered_departments"": [
        6,
        7
      ],
      ""kit_price"": 148
    },
    {
      ""kit_id"": ""S55"",
      ""covered_departments"": [
        13,
        14,
        15
      ],
      ""kit_price"": 141
    },
    {
      ""kit_id"": ""S56"",
      ""covered_departments"": [
        6,
        7,
        8
      ],
      ""kit_price"": 297
    },
    {
      ""kit_id"": ""S57"",
      ""covered_departments"": [
        10,
        11,
        12
      ],
      ""kit_price"": 69
    },
    {
      ""kit_id"": ""S58"",
      ""covered_departments"": [
        1
      ],
      ""kit_price"": 10000
    },
    {
      ""kit_id"": ""S59"",
      ""covered_departments"": [
        2
      ],
      ""kit_price"": 10000
    },
    {
      ""kit_id"": ""S60"",
      ""covered_departments"": [
        3
      ],
      ""kit_price"": 10000
    },
    {
      ""kit_id"": ""S61"",
      ""covered_departments"": [
        4
      ],
      ""kit_price"": 10000
    },
    {
      ""kit_id"": ""S62"",
      ""covered_departments"": [
        5
      ],
      ""kit_price"": 10000
    },
    {
      ""kit_id"": ""S63"",
      ""covered_departments"": [
        6
      ],
      ""kit_price"": 10000
    },
    {
      ""kit_id"": ""S64"",
      ""covered_departments"": [
        7
      ],
      ""kit_price"": 10000
    },
    {
      ""kit_id"": ""S65"",
      ""covered_departments"": [
        8
      ],
      ""kit_price"": 10000
    },
    {
      ""kit_id"": ""S66"",
      ""covered_departments"": [
        9
      ],
      ""kit_price"": 10000
    },
    {
      ""kit_id"": ""S67"",
      ""covered_departments"": [
        10
      ],
      ""kit_price"": 10000
    },
    {
      ""kit_id"": ""S68"",
      ""covered_departments"": [
        11
      ],
      ""kit_price"": 10000
    },
    {
      ""kit_id"": ""S69"",
      ""covered_departments"": [
        12
      ],
      ""kit_price"": 10000
    },
    {
      ""kit_id"": ""S70"",
      ""covered_departments"": [
        13
      ],
      ""kit_price"": 10000
    },
    {
      ""kit_id"": ""S71"",
      ""covered_departments"": [
        14
      ],
      ""kit_price"": 10000
    },
    {
      ""kit_id"": ""S72"",
      ""covered_departments"": [
        15
      ],
      ""kit_price"": 10000
    },
    {
      ""kit_id"": ""S73"",
      ""covered_departments"": [
        16
      ],
      ""kit_price"": 10000
    }
  ]
}

Also, when you send back the chosen kits, please use this simple JSON shape so it's easy to read and process:

{
  ""solution"": [""kit_id"", ...]
}

Here ""solution"" is just the list of kit IDs you want to order — one kit per department — and each string in the array is the identifier for a kit. Think of it like filling out a short form: list the exact box labels you picked. This JSON is just a sketch of the shape I want, not the final answer itself.

Please be sure to use the identifiers exactly as they appear in the instance input — don't rename them or invent new labels.

- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 16, 'num_sets': 73, 'density': 0.1660958904109589, 'sets': [{'id': 1, 'elements': [1, 2, 5, 6], 'cost': 276}, {'id': 2, 'elements': [3, 4, 5, 6, 7], 'cost': 35}, {'id': 3, 'elements': [5, 7, 9], 'cost': 234}, {'id': 4, 'elements': [2, 3, 4], 'cost': 207}, {'id': 5, 'elements': [7, 8], 'cost': 46}, {'id': 6, 'elements': [14, 16], 'cost': 58}, {'id': 7, 'elements': [1, 3, 4, 5], 'cost': 280}, {'id': 8, 'elements': [11, 12], 'cost': 88}, {'id': 9, 'elements': [2, 3], 'cost': 178}, {'id': 10, 'elements': [1, 2], 'cost': 192}, {'id': 11, 'elements': [1, 2, 3, 6], 'cost': 276}, {'id': 12, 'elements': [1, 2, 3, 4], 'cost': 184}, {'id': 13, 'elements': [2, 4, 5], 'cost': 114}, {'id': 14, 'elements': [1, 3], 'cost': 172}, {'id': 15, 'elements': [14, 15, 16], 'cost': 60}, {'id': 16, 'elements': [10, 12], 'cost': 82}, {'id': 17, 'elements': [1, 2, 3], 'cost': 21}, {'id': 18, 'elements': [8, 11], 'cost': 164}, {'id': 19, 'elements': [10, 11, 13], 'cost': 147}, {'id': 20, 'elements': [7, 8, 10], 'cost': 126}, {'id': 21, 'elements': [12, 13], 'cost': 144}, {'id': 22, 'elements': [1, 2, 3, 4, 5], 'cost': 305}, {'id': 23, 'elements': [2, 3, 5], 'cost': 57}, {'id': 24, 'elements': [9, 10, 11, 12], 'cost': 160}, {'id': 25, 'elements': [3, 7, 8], 'cost': 51}, {'id': 26, 'elements': [13, 14, 16], 'cost': 33}, {'id': 27, 'elements': [2, 4, 6], 'cost': 150}, {'id': 28, 'elements': [13, 14], 'cost': 12}, {'id': 29, 'elements': [7, 9], 'cost': 140}, {'id': 30, 'elements': [8, 9, 10], 'cost': 222}, {'id': 31, 'elements': [8, 12], 'cost': 120}, {'id': 32, 'elements': [8, 9, 10, 11, 12], 'cost': 340}, {'id': 33, 'elements': [9, 13], 'cost': 188}, {'id': 34, 'elements': [3, 6], 'cost': 184}, {'id': 35, 'elements': [13, 14, 15, 16], 'cost': 104}, {'id': 36, 'elements': [2, 4], 'cost': 56}, {'id': 37, 'elements': [8, 16], 'cost': 106}, {'id': 38, 'elements': [2, 3, 5, 6], 'cost': 392}, {'id': 39, 'elements': [11, 13, 14, 15, 16], 'cost': 340}, {'id': 40, 'elements': [9, 10, 12], 'cost': 231}, {'id': 41, 'elements': [8, 9, 10, 11], 'cost': 176}, {'id': 42, 'elements': [3, 4, 6, 7], 'cost': 112}, {'id': 43, 'elements': [3, 4, 5, 6], 'cost': 48}, {'id': 44, 'elements': [3, 4], 'cost': 76}, {'id': 45, 'elements': [11, 16], 'cost': 94}, {'id': 46, 'elements': [10, 11, 13, 14], 'cost': 384}, {'id': 47, 'elements': [9, 12, 13, 14], 'cost': 16}, {'id': 48, 'elements': [4, 6, 7, 8, 9], 'cost': 280}, {'id': 49, 'elements': [4, 5, 7, 9], 'cost': 24}, {'id': 50, 'elements': [6, 12], 'cost': 136}, {'id': 51, 'elements': [6, 7, 8, 9, 10], 'cost': 340}, {'id': 52, 'elements': [2, 4, 5, 6], 'cost': 312}, {'id': 53, 'elements': [9, 11, 12, 16], 'cost': 296}, {'id': 54, 'elements': [6, 7], 'cost': 148}, {'id': 55, 'elements': [13, 14, 15], 'cost': 141}, {'id': 56, 'elements': [6, 7, 8], 'cost': 297}, {'id': 57, 'elements': [10, 11, 12], 'cost': 69}, {'id': 58, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 66, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 67, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 68, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 69, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 70, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 71, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 72, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 73, 'elements': [16], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0045_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0045_bag_stats.png'}","[5, 10, 24, 35, 43]",550.0,"{'num_elements': 16, 'num_sets': 73, 'sets': [{'id': 'S1', 'elements': [1, 2, 5, 6], 'cost': 276}, {'id': 'S2', 'elements': [3, 4, 5, 6, 7], 'cost': 35}, {'id': 'S3', 'elements': [5, 7, 9], 'cost': 234}, {'id': 'S4', 'elements': [2, 3, 4], 'cost': 207}, {'id': 'S5', 'elements': [7, 8], 'cost': 46}, {'id': 'S6', 'elements': [14, 16], 'cost': 58}, {'id': 'S7', 'elements': [1, 3, 4, 5], 'cost': 280}, {'id': 'S8', 'elements': [11, 12], 'cost': 88}, {'id': 'S9', 'elements': [2, 3], 'cost': 178}, {'id': 'S10', 'elements': [1, 2], 'cost': 192}, {'id': 'S11', 'elements': [1, 2, 3, 6], 'cost': 276}, {'id': 'S12', 'elements': [1, 2, 3, 4], 'cost': 184}, {'id': 'S13', 'elements': [2, 4, 5], 'cost': 114}, {'id': 'S14', 'elements': [1, 3], 'cost': 172}, {'id': 'S15', 'elements': [14, 15, 16], 'cost': 60}, {'id': 'S16', 'elements': [10, 12], 'cost': 82}, {'id': 'S17', 'elements': [1, 2, 3], 'cost': 21}, {'id': 'S18', 'elements': [8, 11], 'cost': 164}, {'id': 'S19', 'elements': [10, 11, 13], 'cost': 147}, {'id': 'S20', 'elements': [7, 8, 10], 'cost': 126}, {'id': 'S21', 'elements': [12, 13], 'cost': 144}, {'id': 'S22', 'elements': [1, 2, 3, 4, 5], 'cost': 305}, {'id': 'S23', 'elements': [2, 3, 5], 'cost': 57}, {'id': 'S24', 'elements': [9, 10, 11, 12], 'cost': 160}, {'id': 'S25', 'elements': [3, 7, 8], 'cost': 51}, {'id': 'S26', 'elements': [13, 14, 16], 'cost': 33}, {'id': 'S27', 'elements': [2, 4, 6], 'cost': 150}, {'id': 'S28', 'elements': [13, 14], 'cost': 12}, {'id': 'S29', 'elements': [7, 9], 'cost': 140}, {'id': 'S30', 'elements': [8, 9, 10], 'cost': 222}, {'id': 'S31', 'elements': [8, 12], 'cost': 120}, {'id': 'S32', 'elements': [8, 9, 10, 11, 12], 'cost': 340}, {'id': 'S33', 'elements': [9, 13], 'cost': 188}, {'id': 'S34', 'elements': [3, 6], 'cost': 184}, {'id': 'S35', 'elements': [13, 14, 15, 16], 'cost': 104}, {'id': 'S36', 'elements': [2, 4], 'cost': 56}, {'id': 'S37', 'elements': [8, 16], 'cost': 106}, {'id': 'S38', 'elements': [2, 3, 5, 6], 'cost': 392}, {'id': 'S39', 'elements': [11, 13, 14, 15, 16], 'cost': 340}, {'id': 'S40', 'elements': [9, 10, 12], 'cost': 231}, {'id': 'S41', 'elements': [8, 9, 10, 11], 'cost': 176}, {'id': 'S42', 'elements': [3, 4, 6, 7], 'cost': 112}, {'id': 'S43', 'elements': [3, 4, 5, 6], 'cost': 48}, {'id': 'S44', 'elements': [3, 4], 'cost': 76}, {'id': 'S45', 'elements': [11, 16], 'cost': 94}, {'id': 'S46', 'elements': [10, 11, 13, 14], 'cost': 384}, {'id': 'S47', 'elements': [9, 12, 13, 14], 'cost': 16}, {'id': 'S48', 'elements': [4, 6, 7, 8, 9], 'cost': 280}, {'id': 'S49', 'elements': [4, 5, 7, 9], 'cost': 24}, {'id': 'S50', 'elements': [6, 12], 'cost': 136}, {'id': 'S51', 'elements': [6, 7, 8, 9, 10], 'cost': 340}, {'id': 'S52', 'elements': [2, 4, 5, 6], 'cost': 312}, {'id': 'S53', 'elements': [9, 11, 12, 16], 'cost': 296}, {'id': 'S54', 'elements': [6, 7], 'cost': 148}, {'id': 'S55', 'elements': [13, 14, 15], 'cost': 141}, {'id': 'S56', 'elements': [6, 7, 8], 'cost': 297}, {'id': 'S57', 'elements': [10, 11, 12], 'cost': 69}, {'id': 'S58', 'elements': [1], 'cost': 10000}, {'id': 'S59', 'elements': [2], 'cost': 10000}, {'id': 'S60', 'elements': [3], 'cost': 10000}, {'id': 'S61', 'elements': [4], 'cost': 10000}, {'id': 'S62', 'elements': [5], 'cost': 10000}, {'id': 'S63', 'elements': [6], 'cost': 10000}, {'id': 'S64', 'elements': [7], 'cost': 10000}, {'id': 'S65', 'elements': [8], 'cost': 10000}, {'id': 'S66', 'elements': [9], 'cost': 10000}, {'id': 'S67', 'elements': [10], 'cost': 10000}, {'id': 'S68', 'elements': [11], 'cost': 10000}, {'id': 'S69', 'elements': [12], 'cost': 10000}, {'id': 'S70', 'elements': [13], 'cost': 10000}, {'id': 'S71', 'elements': [14], 'cost': 10000}, {'id': 'S72', 'elements': [15], 'cost': 10000}, {'id': 'S73', 'elements': [16], 'cost': 10000}]}","['S5', 'S10', 'S24', 'S35', 'S43']",46,json,1
SPP,SPP,"We’ve got a pile of screening blocks to choose from for the festival, and the job is to pick a combination so each film is scheduled in one—and only one—of the blocks, with no overlap between chosen blocks, while keeping the venue expenses down. Practically speaking, that means selecting blocks that together include every title exactly once, checking that none of the selected blocks repeat a film, and then adding up the rental costs of those blocks to get the total price to minimize. The exact lineup and prices are listed below.

- **total_films**: 14
- **total_blocks_available**: 51

| block_id | block_rental_cost | films_in_block |
|---|---|---|
| S1 | 208 | K L M N |
| S2 | 183 | A C E |
| S3 | 8 | I L |
| S4 | 279 | B D F |
| S5 | 156 | I K |
| S6 | 388 | I J L M |
| S7 | 1 | N |
| S8 | 316 | E F H I |
| S9 | 228 | A B C |
| S10 | 196 | H I J K |
| S11 | 6 | D F H |
| S12 | 57 | L M N |
| S13 | 40 | C E |
| S14 | 70 | M N |
| S15 | 200 | A B C D |
| S16 | 100 | A D |
| S17 | 285 | C D F |
| S18 | 162 | F H |
| S19 | 70 | K M |
| S20 | 77 | H |
| S21 | 36 | A E |
| S22 | 126 | I J |
| S23 | 63 | K M N |
| S24 | 204 | G H K |
| S25 | 90 | C D E |
| S26 | 184 | B E |
| S27 | 44 | B C E G |
| S28 | 42 | B E F |
| S29 | 194 | A B |
| S30 | 174 | H J M |
| S31 | 152 | J K |
| S32 | 26 | D F |
| S33 | 99 | D E I |
| S34 | 256 | E F G H |
| S35 | 246 | A D E |
| S36 | 48 | H J |
| S37 | 108 | C F |
| S38 | 10000 | A |
| S39 | 10000 | B |
| S40 | 10000 | C |
| S41 | 10000 | D |
| S42 | 10000 | E |
| S43 | 10000 | F |
| S44 | 10000 | G |
| S45 | 10000 | H |
| S46 | 10000 | I |
| S47 | 10000 | J |
| S48 | 10000 | K |
| S49 | 10000 | L |
| S50 | 10000 | M |
| S51 | 10000 | N |

Oh, and when you send me the chosen blocks, please use this simple JSON layout so I can read it easily.

{
  ""solution"": [""block_id"", ...]
}

Think of ""solution"" as the list of screening blocks you pick for the lineup — each entry in the array is the identifier for one chosen block (the ""block_id"" in the sketch above). That JSON is just a template to show the shape I expect, not the actual answer itself — replace the placeholders with the real IDs from the instance.

Please be sure to use the exact identifiers from the instance input — do not rename them or invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'problem_type': 'SPP', 'num_elements': 14, 'num_sets': 51, 'density': 0.15826330532212884, 'sets': [{'id': 1, 'elements': [11, 12, 13, 14], 'cost': 208}, {'id': 2, 'elements': [1, 3, 5], 'cost': 183}, {'id': 3, 'elements': [9, 12], 'cost': 8}, {'id': 4, 'elements': [2, 4, 6], 'cost': 279}, {'id': 5, 'elements': [9, 11], 'cost': 156}, {'id': 6, 'elements': [9, 10, 12, 13], 'cost': 388}, {'id': 7, 'elements': [14], 'cost': 1}, {'id': 8, 'elements': [5, 6, 8, 9], 'cost': 316}, {'id': 9, 'elements': [1, 2, 3], 'cost': 228}, {'id': 10, 'elements': [8, 9, 10, 11], 'cost': 196}, {'id': 11, 'elements': [4, 6, 8], 'cost': 6}, {'id': 12, 'elements': [12, 13, 14], 'cost': 57}, {'id': 13, 'elements': [3, 5], 'cost': 40}, {'id': 14, 'elements': [13, 14], 'cost': 70}, {'id': 15, 'elements': [1, 2, 3, 4], 'cost': 200}, {'id': 16, 'elements': [1, 4], 'cost': 100}, {'id': 17, 'elements': [3, 4, 6], 'cost': 285}, {'id': 18, 'elements': [6, 8], 'cost': 162}, {'id': 19, 'elements': [11, 13], 'cost': 70}, {'id': 20, 'elements': [8], 'cost': 77}, {'id': 21, 'elements': [1, 5], 'cost': 36}, {'id': 22, 'elements': [9, 10], 'cost': 126}, {'id': 23, 'elements': [11, 13, 14], 'cost': 63}, {'id': 24, 'elements': [7, 8, 11], 'cost': 204}, {'id': 25, 'elements': [3, 4, 5], 'cost': 90}, {'id': 26, 'elements': [2, 5], 'cost': 184}, {'id': 27, 'elements': [2, 3, 5, 7], 'cost': 44}, {'id': 28, 'elements': [2, 5, 6], 'cost': 42}, {'id': 29, 'elements': [1, 2], 'cost': 194}, {'id': 30, 'elements': [8, 10, 13], 'cost': 174}, {'id': 31, 'elements': [10, 11], 'cost': 152}, {'id': 32, 'elements': [4, 6], 'cost': 26}, {'id': 33, 'elements': [4, 5, 9], 'cost': 99}, {'id': 34, 'elements': [5, 6, 7, 8], 'cost': 256}, {'id': 35, 'elements': [1, 4, 5], 'cost': 246}, {'id': 36, 'elements': [8, 10], 'cost': 48}, {'id': 37, 'elements': [3, 6], 'cost': 108}, {'id': 38, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 39, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 40, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 41, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 42, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 43, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 44, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 45, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 46, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 47, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [14], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0046_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0046_bag_stats.png'}","[3, 14, 16, 18, 27, 31]",536.0,"{'num_elements': 14, 'num_sets': 51, 'sets': [{'id': 'S1', 'elements': ['K', 'L', 'M', 'N'], 'cost': 208}, {'id': 'S2', 'elements': ['A', 'C', 'E'], 'cost': 183}, {'id': 'S3', 'elements': ['I', 'L'], 'cost': 8}, {'id': 'S4', 'elements': ['B', 'D', 'F'], 'cost': 279}, {'id': 'S5', 'elements': ['I', 'K'], 'cost': 156}, {'id': 'S6', 'elements': ['I', 'J', 'L', 'M'], 'cost': 388}, {'id': 'S7', 'elements': ['N'], 'cost': 1}, {'id': 'S8', 'elements': ['E', 'F', 'H', 'I'], 'cost': 316}, {'id': 'S9', 'elements': ['A', 'B', 'C'], 'cost': 228}, {'id': 'S10', 'elements': ['H', 'I', 'J', 'K'], 'cost': 196}, {'id': 'S11', 'elements': ['D', 'F', 'H'], 'cost': 6}, {'id': 'S12', 'elements': ['L', 'M', 'N'], 'cost': 57}, {'id': 'S13', 'elements': ['C', 'E'], 'cost': 40}, {'id': 'S14', 'elements': ['M', 'N'], 'cost': 70}, {'id': 'S15', 'elements': ['A', 'B', 'C', 'D'], 'cost': 200}, {'id': 'S16', 'elements': ['A', 'D'], 'cost': 100}, {'id': 'S17', 'elements': ['C', 'D', 'F'], 'cost': 285}, {'id': 'S18', 'elements': ['F', 'H'], 'cost': 162}, {'id': 'S19', 'elements': ['K', 'M'], 'cost': 70}, {'id': 'S20', 'elements': ['H'], 'cost': 77}, {'id': 'S21', 'elements': ['A', 'E'], 'cost': 36}, {'id': 'S22', 'elements': ['I', 'J'], 'cost': 126}, {'id': 'S23', 'elements': ['K', 'M', 'N'], 'cost': 63}, {'id': 'S24', 'elements': ['G', 'H', 'K'], 'cost': 204}, {'id': 'S25', 'elements': ['C', 'D', 'E'], 'cost': 90}, {'id': 'S26', 'elements': ['B', 'E'], 'cost': 184}, {'id': 'S27', 'elements': ['B', 'C', 'E', 'G'], 'cost': 44}, {'id': 'S28', 'elements': ['B', 'E', 'F'], 'cost': 42}, {'id': 'S29', 'elements': ['A', 'B'], 'cost': 194}, {'id': 'S30', 'elements': ['H', 'J', 'M'], 'cost': 174}, {'id': 'S31', 'elements': ['J', 'K'], 'cost': 152}, {'id': 'S32', 'elements': ['D', 'F'], 'cost': 26}, {'id': 'S33', 'elements': ['D', 'E', 'I'], 'cost': 99}, {'id': 'S34', 'elements': ['E', 'F', 'G', 'H'], 'cost': 256}, {'id': 'S35', 'elements': ['A', 'D', 'E'], 'cost': 246}, {'id': 'S36', 'elements': ['H', 'J'], 'cost': 48}, {'id': 'S37', 'elements': ['C', 'F'], 'cost': 108}, {'id': 'S38', 'elements': ['A'], 'cost': 10000}, {'id': 'S39', 'elements': ['B'], 'cost': 10000}, {'id': 'S40', 'elements': ['C'], 'cost': 10000}, {'id': 'S41', 'elements': ['D'], 'cost': 10000}, {'id': 'S42', 'elements': ['E'], 'cost': 10000}, {'id': 'S43', 'elements': ['F'], 'cost': 10000}, {'id': 'S44', 'elements': ['G'], 'cost': 10000}, {'id': 'S45', 'elements': ['H'], 'cost': 10000}, {'id': 'S46', 'elements': ['I'], 'cost': 10000}, {'id': 'S47', 'elements': ['J'], 'cost': 10000}, {'id': 'S48', 'elements': ['K'], 'cost': 10000}, {'id': 'S49', 'elements': ['L'], 'cost': 10000}, {'id': 'S50', 'elements': ['M'], 'cost': 10000}, {'id': 'S51', 'elements': ['N'], 'cost': 10000}]}","['S3', 'S14', 'S16', 'S18', 'S27', 'S31']",47,markdown_table,names
SPP,SPP,"That Saturday the planting rota required a pick: select seed packs so each garden bed gets seeds from a single chosen pack and the chosen packs don’t overlap on any bed. What makes one selection better than another is the total money spent — just add up the costs of the packs in the selection and aim for the lowest total. The exact beds, pack coverage, and prices are shown below.

- **num_beds**: 14
- **num_packs**: 60

| pack_id | pack_price | beds_covered |
|---|---|---|
| S1 | 156 | 13 14 |
| S2 | 210 | 12 13 14 |
| S3 | 176 | 1 4 |
| S4 | 12 | 4 6 8 10 |
| S5 | 264 | 11 12 13 |
| S6 | 124 | 1 2 3 5 |
| S7 | 4 | 7 8 9 10 |
| S8 | 24 | 10 12 |
| S9 | 232 | 10 11 12 13 |
| S10 | 236 | 1 2 3 4 |
| S11 | 63 | 6 8 10 |
| S12 | 156 | 2 3 |
| S13 | 231 | 4 5 7 |
| S14 | 234 | 3 4 5 |
| S15 | 18 | 12 13 |
| S16 | 291 | 4 5 6 |
| S17 | 280 | 10 11 12 14 |
| S18 | 72 | 7 8 10 |
| S19 | 49 | 10 |
| S20 | 276 | 8 10 11 13 |
| S21 | 195 | 8 10 11 |
| S22 | 123 | 6 9 11 |
| S23 | 93 | 1 2 3 |
| S24 | 94 | 9 11 |
| S25 | 16 | 6 7 8 9 |
| S26 | 132 | 2 4 5 |
| S27 | 104 | 4 5 6 7 |
| S28 | 164 | 12 14 |
| S29 | 70 | 3 4 |
| S30 | 296 | 11 12 13 14 |
| S31 | 96 | 5 6 7 9 |
| S32 | 132 | 4 7 8 |
| S33 | 58 | 9 12 |
| S34 | 260 | 1 3 4 5 |
| S35 | 166 | 11 12 |
| S36 | 9 | 5 6 7 |
| S37 | 174 | 3 5 |
| S38 | 54 | 10 14 |
| S39 | 160 | 1 2 |
| S40 | 220 | 10 12 13 14 |
| S41 | 60 | 4 5 8 |
| S42 | 36 | 10 12 13 |
| S43 | 50 | 5 |
| S44 | 216 | 6 7 9 10 |
| S45 | 364 | 6 7 8 10 |
| S46 | 72 | 8 9 |
| S47 | 10000 | 1 |
| S48 | 10000 | 2 |
| S49 | 10000 | 3 |
| S50 | 10000 | 4 |
| S51 | 10000 | 5 |
| S52 | 10000 | 6 |
| S53 | 10000 | 7 |
| S54 | 10000 | 8 |
| S55 | 10000 | 9 |
| S56 | 10000 | 10 |
| S57 | 10000 | 11 |
| S58 | 10000 | 12 |
| S59 | 10000 | 13 |
| S60 | 10000 | 14 |

When you're ready to give the final pick, just send it in this little JSON shape so it's easy to read and check:

{
  ""solution"": [""pack_id"", ...]
}

This just says ""solution"" with a list of chosen seed pack identifiers (the ""pack_id"" items are placeholders for the real pack IDs from the instance). It's just a sketch of the shape I expect — replace those placeholders with the actual pack IDs when you answer.

Please make sure all identifiers are used exactly as they appear in the instance input — no renaming and no new labels.
- for example: ""1"" or ""23"", ""A"" or ""B"", ""A1"" or ""X7""","{'problem_type': 'SPP', 'num_elements': 14, 'num_sets': 60, 'density': 0.1773809523809524, 'sets': [{'id': 1, 'elements': [13, 14], 'cost': 156}, {'id': 2, 'elements': [12, 13, 14], 'cost': 210}, {'id': 3, 'elements': [1, 4], 'cost': 176}, {'id': 4, 'elements': [4, 6, 8, 10], 'cost': 12}, {'id': 5, 'elements': [11, 12, 13], 'cost': 264}, {'id': 6, 'elements': [1, 2, 3, 5], 'cost': 124}, {'id': 7, 'elements': [7, 8, 9, 10], 'cost': 4}, {'id': 8, 'elements': [10, 12], 'cost': 24}, {'id': 9, 'elements': [10, 11, 12, 13], 'cost': 232}, {'id': 10, 'elements': [1, 2, 3, 4], 'cost': 236}, {'id': 11, 'elements': [6, 8, 10], 'cost': 63}, {'id': 12, 'elements': [2, 3], 'cost': 156}, {'id': 13, 'elements': [4, 5, 7], 'cost': 231}, {'id': 14, 'elements': [3, 4, 5], 'cost': 234}, {'id': 15, 'elements': [12, 13], 'cost': 18}, {'id': 16, 'elements': [4, 5, 6], 'cost': 291}, {'id': 17, 'elements': [10, 11, 12, 14], 'cost': 280}, {'id': 18, 'elements': [7, 8, 10], 'cost': 72}, {'id': 19, 'elements': [10], 'cost': 49}, {'id': 20, 'elements': [8, 10, 11, 13], 'cost': 276}, {'id': 21, 'elements': [8, 10, 11], 'cost': 195}, {'id': 22, 'elements': [6, 9, 11], 'cost': 123}, {'id': 23, 'elements': [1, 2, 3], 'cost': 93}, {'id': 24, 'elements': [9, 11], 'cost': 94}, {'id': 25, 'elements': [6, 7, 8, 9], 'cost': 16}, {'id': 26, 'elements': [2, 4, 5], 'cost': 132}, {'id': 27, 'elements': [4, 5, 6, 7], 'cost': 104}, {'id': 28, 'elements': [12, 14], 'cost': 164}, {'id': 29, 'elements': [3, 4], 'cost': 70}, {'id': 30, 'elements': [11, 12, 13, 14], 'cost': 296}, {'id': 31, 'elements': [5, 6, 7, 9], 'cost': 96}, {'id': 32, 'elements': [4, 7, 8], 'cost': 132}, {'id': 33, 'elements': [9, 12], 'cost': 58}, {'id': 34, 'elements': [1, 3, 4, 5], 'cost': 260}, {'id': 35, 'elements': [11, 12], 'cost': 166}, {'id': 36, 'elements': [5, 6, 7], 'cost': 9}, {'id': 37, 'elements': [3, 5], 'cost': 174}, {'id': 38, 'elements': [10, 14], 'cost': 54}, {'id': 39, 'elements': [1, 2], 'cost': 160}, {'id': 40, 'elements': [10, 12, 13, 14], 'cost': 220}, {'id': 41, 'elements': [4, 5, 8], 'cost': 60}, {'id': 42, 'elements': [10, 12, 13], 'cost': 36}, {'id': 43, 'elements': [5], 'cost': 50}, {'id': 44, 'elements': [6, 7, 9, 10], 'cost': 216}, {'id': 45, 'elements': [6, 7, 8, 10], 'cost': 364}, {'id': 46, 'elements': [8, 9], 'cost': 72}, {'id': 47, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [14], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0047_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0047_bag_stats.png'}","[6, 15, 22, 32, 38]",451.0,"{'num_elements': 14, 'num_sets': 60, 'sets': [{'id': 'S1', 'elements': [13, 14], 'cost': 156}, {'id': 'S2', 'elements': [12, 13, 14], 'cost': 210}, {'id': 'S3', 'elements': [1, 4], 'cost': 176}, {'id': 'S4', 'elements': [4, 6, 8, 10], 'cost': 12}, {'id': 'S5', 'elements': [11, 12, 13], 'cost': 264}, {'id': 'S6', 'elements': [1, 2, 3, 5], 'cost': 124}, {'id': 'S7', 'elements': [7, 8, 9, 10], 'cost': 4}, {'id': 'S8', 'elements': [10, 12], 'cost': 24}, {'id': 'S9', 'elements': [10, 11, 12, 13], 'cost': 232}, {'id': 'S10', 'elements': [1, 2, 3, 4], 'cost': 236}, {'id': 'S11', 'elements': [6, 8, 10], 'cost': 63}, {'id': 'S12', 'elements': [2, 3], 'cost': 156}, {'id': 'S13', 'elements': [4, 5, 7], 'cost': 231}, {'id': 'S14', 'elements': [3, 4, 5], 'cost': 234}, {'id': 'S15', 'elements': [12, 13], 'cost': 18}, {'id': 'S16', 'elements': [4, 5, 6], 'cost': 291}, {'id': 'S17', 'elements': [10, 11, 12, 14], 'cost': 280}, {'id': 'S18', 'elements': [7, 8, 10], 'cost': 72}, {'id': 'S19', 'elements': [10], 'cost': 49}, {'id': 'S20', 'elements': [8, 10, 11, 13], 'cost': 276}, {'id': 'S21', 'elements': [8, 10, 11], 'cost': 195}, {'id': 'S22', 'elements': [6, 9, 11], 'cost': 123}, {'id': 'S23', 'elements': [1, 2, 3], 'cost': 93}, {'id': 'S24', 'elements': [9, 11], 'cost': 94}, {'id': 'S25', 'elements': [6, 7, 8, 9], 'cost': 16}, {'id': 'S26', 'elements': [2, 4, 5], 'cost': 132}, {'id': 'S27', 'elements': [4, 5, 6, 7], 'cost': 104}, {'id': 'S28', 'elements': [12, 14], 'cost': 164}, {'id': 'S29', 'elements': [3, 4], 'cost': 70}, {'id': 'S30', 'elements': [11, 12, 13, 14], 'cost': 296}, {'id': 'S31', 'elements': [5, 6, 7, 9], 'cost': 96}, {'id': 'S32', 'elements': [4, 7, 8], 'cost': 132}, {'id': 'S33', 'elements': [9, 12], 'cost': 58}, {'id': 'S34', 'elements': [1, 3, 4, 5], 'cost': 260}, {'id': 'S35', 'elements': [11, 12], 'cost': 166}, {'id': 'S36', 'elements': [5, 6, 7], 'cost': 9}, {'id': 'S37', 'elements': [3, 5], 'cost': 174}, {'id': 'S38', 'elements': [10, 14], 'cost': 54}, {'id': 'S39', 'elements': [1, 2], 'cost': 160}, {'id': 'S40', 'elements': [10, 12, 13, 14], 'cost': 220}, {'id': 'S41', 'elements': [4, 5, 8], 'cost': 60}, {'id': 'S42', 'elements': [10, 12, 13], 'cost': 36}, {'id': 'S43', 'elements': [5], 'cost': 50}, {'id': 'S44', 'elements': [6, 7, 9, 10], 'cost': 216}, {'id': 'S45', 'elements': [6, 7, 8, 10], 'cost': 364}, {'id': 'S46', 'elements': [8, 9], 'cost': 72}, {'id': 'S47', 'elements': [1], 'cost': 10000}, {'id': 'S48', 'elements': [2], 'cost': 10000}, {'id': 'S49', 'elements': [3], 'cost': 10000}, {'id': 'S50', 'elements': [4], 'cost': 10000}, {'id': 'S51', 'elements': [5], 'cost': 10000}, {'id': 'S52', 'elements': [6], 'cost': 10000}, {'id': 'S53', 'elements': [7], 'cost': 10000}, {'id': 'S54', 'elements': [8], 'cost': 10000}, {'id': 'S55', 'elements': [9], 'cost': 10000}, {'id': 'S56', 'elements': [10], 'cost': 10000}, {'id': 'S57', 'elements': [11], 'cost': 10000}, {'id': 'S58', 'elements': [12], 'cost': 10000}, {'id': 'S59', 'elements': [13], 'cost': 10000}, {'id': 'S60', 'elements': [14], 'cost': 10000}]}","['S6', 'S15', 'S22', 'S32', 'S38']",48,markdown_table,1
SPP,SPP,"Many people on the editorial side are weighing which bundles to publish so each interest area in the subscriber base is covered once and only once, and no two selected bundles include the same interest. What counts as a better selection is a lower editing bill: just total the editing charges for the chosen bundles to see which option is cheapest. The specific bundle choices, their topics, and associated costs are shown below.

{
  ""total_interest_areas"": 17,
  ""total_bundle_options"": 79,
  ""sets"": [
    {
      ""bundle_id"": ""S1"",
      ""bundle_interest_areas"": [
        12,
        13,
        14,
        16
      ],
      ""bundle_cost"": 316
    },
    {
      ""bundle_id"": ""S2"",
      ""bundle_interest_areas"": [
        14,
        15,
        16
      ],
      ""bundle_cost"": 33
    },
    {
      ""bundle_id"": ""S3"",
      ""bundle_interest_areas"": [
        9,
        10,
        15,
        16
      ],
      ""bundle_cost"": 24
    },
    {
      ""bundle_id"": ""S4"",
      ""bundle_interest_areas"": [
        7,
        8
      ],
      ""bundle_cost"": 166
    },
    {
      ""bundle_id"": ""S5"",
      ""bundle_interest_areas"": [
        0,
        2,
        3,
        4,
        5
      ],
      ""bundle_cost"": 210
    },
    {
      ""bundle_id"": ""S6"",
      ""bundle_interest_areas"": [
        0,
        1,
        2,
        3,
        4
      ],
      ""bundle_cost"": 275
    },
    {
      ""bundle_id"": ""S7"",
      ""bundle_interest_areas"": [
        3,
        4,
        5,
        6
      ],
      ""bundle_cost"": 316
    },
    {
      ""bundle_id"": ""S8"",
      ""bundle_interest_areas"": [
        13,
        15,
        16
      ],
      ""bundle_cost"": 87
    },
    {
      ""bundle_id"": ""S9"",
      ""bundle_interest_areas"": [
        13,
        14,
        15
      ],
      ""bundle_cost"": 288
    },
    {
      ""bundle_id"": ""S10"",
      ""bundle_interest_areas"": [
        13,
        15
      ],
      ""bundle_cost"": 182
    },
    {
      ""bundle_id"": ""S11"",
      ""bundle_interest_areas"": [
        10,
        14
      ],
      ""bundle_cost"": 18
    },
    {
      ""bundle_id"": ""S12"",
      ""bundle_interest_areas"": [
        3,
        6,
        7
      ],
      ""bundle_cost"": 54
    },
    {
      ""bundle_id"": ""S13"",
      ""bundle_interest_areas"": [
        1,
        3
      ],
      ""bundle_cost"": 166
    },
    {
      ""bundle_id"": ""S14"",
      ""bundle_interest_areas"": [
        6,
        10
      ],
      ""bundle_cost"": 190
    },
    {
      ""bundle_id"": ""S15"",
      ""bundle_interest_areas"": [
        1,
        2
      ],
      ""bundle_cost"": 126
    },
    {
      ""bundle_id"": ""S16"",
      ""bundle_interest_areas"": [
        13,
        14
      ],
      ""bundle_cost"": 198
    },
    {
      ""bundle_id"": ""S17"",
      ""bundle_interest_areas"": [
        10,
        11
      ],
      ""bundle_cost"": 192
    },
    {
      ""bundle_id"": ""S18"",
      ""bundle_interest_areas"": [
        6,
        7,
        8,
        10,
        11
      ],
      ""bundle_cost"": 400
    },
    {
      ""bundle_id"": ""S19"",
      ""bundle_interest_areas"": [
        1,
        3,
        4
      ],
      ""bundle_cost"": 171
    },
    {
      ""bundle_id"": ""S20"",
      ""bundle_interest_areas"": [
        12,
        13,
        14,
        15,
        16
      ],
      ""bundle_cost"": 140
    },
    {
      ""bundle_id"": ""S21"",
      ""bundle_interest_areas"": [
        3,
        4,
        5,
        7,
        8
      ],
      ""bundle_cost"": 365
    },
    {
      ""bundle_id"": ""S22"",
      ""bundle_interest_areas"": [
        5,
        6,
        7,
        8,
        9
      ],
      ""bundle_cost"": 425
    },
    {
      ""bundle_id"": ""S23"",
      ""bundle_interest_areas"": [
        5,
        6,
        8,
        9
      ],
      ""bundle_cost"": 160
    },
    {
      ""bundle_id"": ""S24"",
      ""bundle_interest_areas"": [
        3,
        4,
        5
      ],
      ""bundle_cost"": 192
    },
    {
      ""bundle_id"": ""S25"",
      ""bundle_interest_areas"": [
        11,
        15
      ],
      ""bundle_cost"": 148
    },
    {
      ""bundle_id"": ""S26"",
      ""bundle_interest_areas"": [
        7,
        8,
        11
      ],
      ""bundle_cost"": 276
    },
    {
      ""bundle_id"": ""S27"",
      ""bundle_interest_areas"": [
        4,
        5,
        6,
        7
      ],
      ""bundle_cost"": 4
    },
    {
      ""bundle_id"": ""S28"",
      ""bundle_interest_areas"": [
        3,
        5
      ],
      ""bundle_cost"": 86
    },
    {
      ""bundle_id"": ""S29"",
      ""bundle_interest_areas"": [
        8,
        13
      ],
      ""bundle_cost"": 134
    },
    {
      ""bundle_id"": ""S30"",
      ""bundle_interest_areas"": [
        1,
        5,
        7
      ],
      ""bundle_cost"": 54
    },
    {
      ""bundle_id"": ""S31"",
      ""bundle_interest_areas"": [
        9,
        10,
        12,
        13
      ],
      ""bundle_cost"": 20
    },
    {
      ""bundle_id"": ""S32"",
      ""bundle_interest_areas"": [
        6,
        7,
        8,
        9,
        10
      ],
      ""bundle_cost"": 325
    },
    {
      ""bundle_id"": ""S33"",
      ""bundle_interest_areas"": [
        12,
        13,
        15
      ],
      ""bundle_cost"": 273
    },
    {
      ""bundle_id"": ""S34"",
      ""bundle_interest_areas"": [
        0,
        1,
        2,
        3,
        5
      ],
      ""bundle_cost"": 370
    },
    {
      ""bundle_id"": ""S35"",
      ""bundle_interest_areas"": [
        7,
        8,
        9,
        10,
        12
      ],
      ""bundle_cost"": 470
    },
    {
      ""bundle_id"": ""S36"",
      ""bundle_interest_areas"": [
        0,
        1,
        3,
        4
      ],
      ""bundle_cost"": 376
    },
    {
      ""bundle_id"": ""S37"",
      ""bundle_interest_areas"": [
        9,
        10,
        11,
        12,
        14
      ],
      ""bundle_cost"": 330
    },
    {
      ""bundle_id"": ""S38"",
      ""bundle_interest_areas"": [
        6,
        8
      ],
      ""bundle_cost"": 122
    },
    {
      ""bundle_id"": ""S39"",
      ""bundle_interest_areas"": [
        13,
        16
      ],
      ""bundle_cost"": 38
    },
    {
      ""bundle_id"": ""S40"",
      ""bundle_interest_areas"": [
        2,
        3,
        5
      ],
      ""bundle_cost"": 264
    },
    {
      ""bundle_id"": ""S41"",
      ""bundle_interest_areas"": [
        11,
        12,
        13,
        15,
        16
      ],
      ""bundle_cost"": 435
    },
    {
      ""bundle_id"": ""S42"",
      ""bundle_interest_areas"": [
        10,
        12,
        14,
        15,
        16
      ],
      ""bundle_cost"": 370
    },
    {
      ""bundle_id"": ""S43"",
      ""bundle_interest_areas"": [
        11,
        13
      ],
      ""bundle_cost"": 170
    },
    {
      ""bundle_id"": ""S44"",
      ""bundle_interest_areas"": [
        7,
        10
      ],
      ""bundle_cost"": 196
    },
    {
      ""bundle_id"": ""S45"",
      ""bundle_interest_areas"": [
        3,
        4,
        6,
        7
      ],
      ""bundle_cost"": 324
    },
    {
      ""bundle_id"": ""S46"",
      ""bundle_interest_areas"": [
        9,
        10,
        12
      ],
      ""bundle_cost"": 54
    },
    {
      ""bundle_id"": ""S47"",
      ""bundle_interest_areas"": [
        5,
        8
      ],
      ""bundle_cost"": 152
    },
    {
      ""bundle_id"": ""S48"",
      ""bundle_interest_areas"": [
        10,
        15
      ],
      ""bundle_cost"": 98
    },
    {
      ""bundle_id"": ""S49"",
      ""bundle_interest_areas"": [
        7,
        9,
        11,
        12
      ],
      ""bundle_cost"": 52
    },
    {
      ""bundle_id"": ""S50"",
      ""bundle_interest_areas"": [
        12,
        13
      ],
      ""bundle_cost"": 110
    },
    {
      ""bundle_id"": ""S51"",
      ""bundle_interest_areas"": [
        7,
        9,
        10,
        12,
        13
      ],
      ""bundle_cost"": 170
    },
    {
      ""bundle_id"": ""S52"",
      ""bundle_interest_areas"": [
        1,
        3,
        4,
        6,
        8
      ],
      ""bundle_cost"": 105
    },
    {
      ""bundle_id"": ""S53"",
      ""bundle_interest_areas"": [
        13,
        14,
        16
      ],
      ""bundle_cost"": 171
    },
    {
      ""bundle_id"": ""S54"",
      ""bundle_interest_areas"": [
        0,
        1,
        2,
        4
      ],
      ""bundle_cost"": 68
    },
    {
      ""bundle_id"": ""S55"",
      ""bundle_interest_areas"": [
        11,
        12,
        13,
        15
      ],
      ""bundle_cost"": 220
    },
    {
      ""bundle_id"": ""S56"",
      ""bundle_interest_areas"": [
        7,
        8,
        9,
        10,
        11
      ],
      ""bundle_cost"": 185
    },
    {
      ""bundle_id"": ""S57"",
      ""bundle_interest_areas"": [
        4,
        7,
        8
      ],
      ""bundle_cost"": 54
    },
    {
      ""bundle_id"": ""S58"",
      ""bundle_interest_areas"": [
        3,
        5,
        6,
        7,
        8
      ],
      ""bundle_cost"": 330
    },
    {
      ""bundle_id"": ""S59"",
      ""bundle_interest_areas"": [
        3,
        8
      ],
      ""bundle_cost"": 106
    },
    {
      ""bundle_id"": ""S60"",
      ""bundle_interest_areas"": [
        2,
        3,
        4,
        6
      ],
      ""bundle_cost"": 220
    },
    {
      ""bundle_id"": ""S61"",
      ""bundle_interest_areas"": [
        0,
        1,
        2
      ],
      ""bundle_cost"": 174
    },
    {
      ""bundle_id"": ""S62"",
      ""bundle_interest_areas"": [
        0,
        2
      ],
      ""bundle_cost"": 88
    },
    {
      ""bundle_id"": ""S63"",
      ""bundle_interest_areas"": [
        0
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S64"",
      ""bundle_interest_areas"": [
        1
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S65"",
      ""bundle_interest_areas"": [
        2
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S66"",
      ""bundle_interest_areas"": [
        3
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S67"",
      ""bundle_interest_areas"": [
        4
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S68"",
      ""bundle_interest_areas"": [
        5
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S69"",
      ""bundle_interest_areas"": [
        6
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S70"",
      ""bundle_interest_areas"": [
        7
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S71"",
      ""bundle_interest_areas"": [
        8
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S72"",
      ""bundle_interest_areas"": [
        9
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S73"",
      ""bundle_interest_areas"": [
        10
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S74"",
      ""bundle_interest_areas"": [
        11
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S75"",
      ""bundle_interest_areas"": [
        12
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S76"",
      ""bundle_interest_areas"": [
        13
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S77"",
      ""bundle_interest_areas"": [
        14
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S78"",
      ""bundle_interest_areas"": [
        15
      ],
      ""bundle_cost"": 10000
    },
    {
      ""bundle_id"": ""S79"",
      ""bundle_interest_areas"": [
        16
      ],
      ""bundle_cost"": 10000
    }
  ]
}

If you want, reply using this simple JSON shape so it’s easy to read and machine-friendly:

{
  ""solution"": [""bundle_id""]
}

Here ""solution"" is just the list of bundle ids you’d choose to publish — put each bundle’s exact id string in that array. Think of it like filling out a tiny form: the array lists the chosen bundles, and that’s all it needs to show. This is only a sketch of the expected shape, not the actual answer.

Please use the identifiers exactly as they appear in the instance input — no renaming and no new labels.
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 17, 'num_sets': 79, 'density': 0.1690245718540581, 'sets': [{'id': 1, 'elements': [13, 14, 15, 17], 'cost': 316}, {'id': 2, 'elements': [15, 16, 17], 'cost': 33}, {'id': 3, 'elements': [10, 11, 16, 17], 'cost': 24}, {'id': 4, 'elements': [8, 9], 'cost': 166}, {'id': 5, 'elements': [1, 3, 4, 5, 6], 'cost': 210}, {'id': 6, 'elements': [1, 2, 3, 4, 5], 'cost': 275}, {'id': 7, 'elements': [4, 5, 6, 7], 'cost': 316}, {'id': 8, 'elements': [14, 16, 17], 'cost': 87}, {'id': 9, 'elements': [14, 15, 16], 'cost': 288}, {'id': 10, 'elements': [14, 16], 'cost': 182}, {'id': 11, 'elements': [11, 15], 'cost': 18}, {'id': 12, 'elements': [4, 7, 8], 'cost': 54}, {'id': 13, 'elements': [2, 4], 'cost': 166}, {'id': 14, 'elements': [7, 11], 'cost': 190}, {'id': 15, 'elements': [2, 3], 'cost': 126}, {'id': 16, 'elements': [14, 15], 'cost': 198}, {'id': 17, 'elements': [11, 12], 'cost': 192}, {'id': 18, 'elements': [7, 8, 9, 11, 12], 'cost': 400}, {'id': 19, 'elements': [2, 4, 5], 'cost': 171}, {'id': 20, 'elements': [13, 14, 15, 16, 17], 'cost': 140}, {'id': 21, 'elements': [4, 5, 6, 8, 9], 'cost': 365}, {'id': 22, 'elements': [6, 7, 8, 9, 10], 'cost': 425}, {'id': 23, 'elements': [6, 7, 9, 10], 'cost': 160}, {'id': 24, 'elements': [4, 5, 6], 'cost': 192}, {'id': 25, 'elements': [12, 16], 'cost': 148}, {'id': 26, 'elements': [8, 9, 12], 'cost': 276}, {'id': 27, 'elements': [5, 6, 7, 8], 'cost': 4}, {'id': 28, 'elements': [4, 6], 'cost': 86}, {'id': 29, 'elements': [9, 14], 'cost': 134}, {'id': 30, 'elements': [2, 6, 8], 'cost': 54}, {'id': 31, 'elements': [10, 11, 13, 14], 'cost': 20}, {'id': 32, 'elements': [7, 8, 9, 10, 11], 'cost': 325}, {'id': 33, 'elements': [13, 14, 16], 'cost': 273}, {'id': 34, 'elements': [1, 2, 3, 4, 6], 'cost': 370}, {'id': 35, 'elements': [8, 9, 10, 11, 13], 'cost': 470}, {'id': 36, 'elements': [1, 2, 4, 5], 'cost': 376}, {'id': 37, 'elements': [10, 11, 12, 13, 15], 'cost': 330}, {'id': 38, 'elements': [7, 9], 'cost': 122}, {'id': 39, 'elements': [14, 17], 'cost': 38}, {'id': 40, 'elements': [3, 4, 6], 'cost': 264}, {'id': 41, 'elements': [12, 13, 14, 16, 17], 'cost': 435}, {'id': 42, 'elements': [11, 13, 15, 16, 17], 'cost': 370}, {'id': 43, 'elements': [12, 14], 'cost': 170}, {'id': 44, 'elements': [8, 11], 'cost': 196}, {'id': 45, 'elements': [4, 5, 7, 8], 'cost': 324}, {'id': 46, 'elements': [10, 11, 13], 'cost': 54}, {'id': 47, 'elements': [6, 9], 'cost': 152}, {'id': 48, 'elements': [11, 16], 'cost': 98}, {'id': 49, 'elements': [8, 10, 12, 13], 'cost': 52}, {'id': 50, 'elements': [13, 14], 'cost': 110}, {'id': 51, 'elements': [8, 10, 11, 13, 14], 'cost': 170}, {'id': 52, 'elements': [2, 4, 5, 7, 9], 'cost': 105}, {'id': 53, 'elements': [14, 15, 17], 'cost': 171}, {'id': 54, 'elements': [1, 2, 3, 5], 'cost': 68}, {'id': 55, 'elements': [12, 13, 14, 16], 'cost': 220}, {'id': 56, 'elements': [8, 9, 10, 11, 12], 'cost': 185}, {'id': 57, 'elements': [5, 8, 9], 'cost': 54}, {'id': 58, 'elements': [4, 6, 7, 8, 9], 'cost': 330}, {'id': 59, 'elements': [4, 9], 'cost': 106}, {'id': 60, 'elements': [3, 4, 5, 7], 'cost': 220}, {'id': 61, 'elements': [1, 2, 3], 'cost': 174}, {'id': 62, 'elements': [1, 3], 'cost': 88}, {'id': 63, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 65, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 66, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 67, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 68, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 69, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 70, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 71, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 72, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 73, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 74, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 75, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 76, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 77, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 78, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 79, 'elements': [17], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0048_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0048_bag_stats.png'}","[8, 11, 28, 38, 49, 54]",433.0,"{'num_elements': 17, 'num_sets': 79, 'sets': [{'id': 'S1', 'elements': [12, 13, 14, 16], 'cost': 316}, {'id': 'S2', 'elements': [14, 15, 16], 'cost': 33}, {'id': 'S3', 'elements': [9, 10, 15, 16], 'cost': 24}, {'id': 'S4', 'elements': [7, 8], 'cost': 166}, {'id': 'S5', 'elements': [0, 2, 3, 4, 5], 'cost': 210}, {'id': 'S6', 'elements': [0, 1, 2, 3, 4], 'cost': 275}, {'id': 'S7', 'elements': [3, 4, 5, 6], 'cost': 316}, {'id': 'S8', 'elements': [13, 15, 16], 'cost': 87}, {'id': 'S9', 'elements': [13, 14, 15], 'cost': 288}, {'id': 'S10', 'elements': [13, 15], 'cost': 182}, {'id': 'S11', 'elements': [10, 14], 'cost': 18}, {'id': 'S12', 'elements': [3, 6, 7], 'cost': 54}, {'id': 'S13', 'elements': [1, 3], 'cost': 166}, {'id': 'S14', 'elements': [6, 10], 'cost': 190}, {'id': 'S15', 'elements': [1, 2], 'cost': 126}, {'id': 'S16', 'elements': [13, 14], 'cost': 198}, {'id': 'S17', 'elements': [10, 11], 'cost': 192}, {'id': 'S18', 'elements': [6, 7, 8, 10, 11], 'cost': 400}, {'id': 'S19', 'elements': [1, 3, 4], 'cost': 171}, {'id': 'S20', 'elements': [12, 13, 14, 15, 16], 'cost': 140}, {'id': 'S21', 'elements': [3, 4, 5, 7, 8], 'cost': 365}, {'id': 'S22', 'elements': [5, 6, 7, 8, 9], 'cost': 425}, {'id': 'S23', 'elements': [5, 6, 8, 9], 'cost': 160}, {'id': 'S24', 'elements': [3, 4, 5], 'cost': 192}, {'id': 'S25', 'elements': [11, 15], 'cost': 148}, {'id': 'S26', 'elements': [7, 8, 11], 'cost': 276}, {'id': 'S27', 'elements': [4, 5, 6, 7], 'cost': 4}, {'id': 'S28', 'elements': [3, 5], 'cost': 86}, {'id': 'S29', 'elements': [8, 13], 'cost': 134}, {'id': 'S30', 'elements': [1, 5, 7], 'cost': 54}, {'id': 'S31', 'elements': [9, 10, 12, 13], 'cost': 20}, {'id': 'S32', 'elements': [6, 7, 8, 9, 10], 'cost': 325}, {'id': 'S33', 'elements': [12, 13, 15], 'cost': 273}, {'id': 'S34', 'elements': [0, 1, 2, 3, 5], 'cost': 370}, {'id': 'S35', 'elements': [7, 8, 9, 10, 12], 'cost': 470}, {'id': 'S36', 'elements': [0, 1, 3, 4], 'cost': 376}, {'id': 'S37', 'elements': [9, 10, 11, 12, 14], 'cost': 330}, {'id': 'S38', 'elements': [6, 8], 'cost': 122}, {'id': 'S39', 'elements': [13, 16], 'cost': 38}, {'id': 'S40', 'elements': [2, 3, 5], 'cost': 264}, {'id': 'S41', 'elements': [11, 12, 13, 15, 16], 'cost': 435}, {'id': 'S42', 'elements': [10, 12, 14, 15, 16], 'cost': 370}, {'id': 'S43', 'elements': [11, 13], 'cost': 170}, {'id': 'S44', 'elements': [7, 10], 'cost': 196}, {'id': 'S45', 'elements': [3, 4, 6, 7], 'cost': 324}, {'id': 'S46', 'elements': [9, 10, 12], 'cost': 54}, {'id': 'S47', 'elements': [5, 8], 'cost': 152}, {'id': 'S48', 'elements': [10, 15], 'cost': 98}, {'id': 'S49', 'elements': [7, 9, 11, 12], 'cost': 52}, {'id': 'S50', 'elements': [12, 13], 'cost': 110}, {'id': 'S51', 'elements': [7, 9, 10, 12, 13], 'cost': 170}, {'id': 'S52', 'elements': [1, 3, 4, 6, 8], 'cost': 105}, {'id': 'S53', 'elements': [13, 14, 16], 'cost': 171}, {'id': 'S54', 'elements': [0, 1, 2, 4], 'cost': 68}, {'id': 'S55', 'elements': [11, 12, 13, 15], 'cost': 220}, {'id': 'S56', 'elements': [7, 8, 9, 10, 11], 'cost': 185}, {'id': 'S57', 'elements': [4, 7, 8], 'cost': 54}, {'id': 'S58', 'elements': [3, 5, 6, 7, 8], 'cost': 330}, {'id': 'S59', 'elements': [3, 8], 'cost': 106}, {'id': 'S60', 'elements': [2, 3, 4, 6], 'cost': 220}, {'id': 'S61', 'elements': [0, 1, 2], 'cost': 174}, {'id': 'S62', 'elements': [0, 2], 'cost': 88}, {'id': 'S63', 'elements': [0], 'cost': 10000}, {'id': 'S64', 'elements': [1], 'cost': 10000}, {'id': 'S65', 'elements': [2], 'cost': 10000}, {'id': 'S66', 'elements': [3], 'cost': 10000}, {'id': 'S67', 'elements': [4], 'cost': 10000}, {'id': 'S68', 'elements': [5], 'cost': 10000}, {'id': 'S69', 'elements': [6], 'cost': 10000}, {'id': 'S70', 'elements': [7], 'cost': 10000}, {'id': 'S71', 'elements': [8], 'cost': 10000}, {'id': 'S72', 'elements': [9], 'cost': 10000}, {'id': 'S73', 'elements': [10], 'cost': 10000}, {'id': 'S74', 'elements': [11], 'cost': 10000}, {'id': 'S75', 'elements': [12], 'cost': 10000}, {'id': 'S76', 'elements': [13], 'cost': 10000}, {'id': 'S77', 'elements': [14], 'cost': 10000}, {'id': 'S78', 'elements': [15], 'cost': 10000}, {'id': 'S79', 'elements': [16], 'cost': 10000}]}","['S8', 'S11', 'S28', 'S38', 'S49', 'S54']",49,json,0
SPP,SPP,"Many sysadmins face the same tidy-up: choose which backup archives to keep so every file is stored in one chosen archive and nowhere else, and don’t let any file fall through the cracks. The practical test is monetary — total storage cost equals the sum of costs for the kept archives, and the aim is to make that sum as small as possible. The concrete backup list, file coverage, and costs appear below.

- **total_files**: 18
- **total_archives**: 64

| archive_id | archive_cost | files_in_archive |
|---|---|---|
| S1 | 64 | 14 17 |
| S2 | 74 | 7 11 |
| S3 | 156 | 15 16 18 |
| S4 | 48 | 4 7 |
| S5 | 231 | 16 17 18 |
| S6 | 258 | 1 2 3 |
| S7 | 280 | 5 7 8 9 10 |
| S8 | 450 | 11 13 14 16 17 |
| S9 | 215 | 1 2 3 4 5 |
| S10 | 344 | 1 2 3 4 |
| S11 | 93 | 6 8 10 |
| S12 | 324 | 15 16 17 18 |
| S13 | 152 | 9 10 |
| S14 | 245 | 6 8 9 10 11 |
| S15 | 252 | 8 9 12 13 |
| S16 | 190 | 9 10 11 12 13 |
| S17 | 136 | 1 4 5 9 |
| S18 | 36 | 16 17 |
| S19 | 140 | 12 15 |
| S20 | 155 | 7 9 10 11 12 |
| S21 | 9 | 2 5 7 |
| S22 | 68 | 5 6 |
| S23 | 138 | 3 4 8 |
| S24 | 8 | 6 7 8 9 |
| S25 | 156 | 1 2 4 5 |
| S26 | 256 | 2 3 4 6 |
| S27 | 80 | 8 9 10 12 |
| S28 | 186 | 13 16 18 |
| S29 | 415 | 1 2 3 5 6 |
| S30 | 216 | 15 17 18 |
| S31 | 255 | 12 13 15 |
| S32 | 360 | 11 14 15 16 |
| S33 | 16 | 11 12 13 15 |
| S34 | 216 | 13 14 16 |
| S35 | 294 | 11 13 17 |
| S36 | 222 | 8 13 15 |
| S37 | 280 | 4 6 7 8 10 |
| S38 | 194 | 7 9 |
| S39 | 138 | 4 5 6 |
| S40 | 112 | 3 7 8 11 |
| S41 | 132 | 11 13 16 18 |
| S42 | 9 | 14 16 18 |
| S43 | 65 | 9 10 11 12 14 |
| S44 | 219 | 1 3 4 |
| S45 | 288 | 3 5 6 |
| S46 | 138 | 15 16 |
| S47 | 10000 | 1 |
| S48 | 10000 | 2 |
| S49 | 10000 | 3 |
| S50 | 10000 | 4 |
| S51 | 10000 | 5 |
| S52 | 10000 | 6 |
| S53 | 10000 | 7 |
| S54 | 10000 | 8 |
| S55 | 10000 | 9 |
| S56 | 10000 | 10 |
| S57 | 10000 | 11 |
| S58 | 10000 | 12 |
| S59 | 10000 | 13 |
| S60 | 10000 | 14 |
| S61 | 10000 | 15 |
| S62 | 10000 | 16 |
| S63 | 10000 | 17 |
| S64 | 10000 | 18 |

Oh, and when you reply, just drop the chosen archive IDs into a tiny JSON outline like this (super simple):

{
  ""solution"": [""archive_id"", ...]
}

Think of ""solution"" as the list of archives you decide to keep — each string in that array is an archive identifier placeholder. This is just a sketch of the shape I expect, not the actual answer.

Please make sure you use the exact identifiers from the instance input — no renaming and no new labels.  
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'problem_type': 'SPP', 'num_elements': 18, 'num_sets': 64, 'density': 0.15364583333333334, 'sets': [{'id': 1, 'elements': [14, 17], 'cost': 64}, {'id': 2, 'elements': [7, 11], 'cost': 74}, {'id': 3, 'elements': [15, 16, 18], 'cost': 156}, {'id': 4, 'elements': [4, 7], 'cost': 48}, {'id': 5, 'elements': [16, 17, 18], 'cost': 231}, {'id': 6, 'elements': [1, 2, 3], 'cost': 258}, {'id': 7, 'elements': [5, 7, 8, 9, 10], 'cost': 280}, {'id': 8, 'elements': [11, 13, 14, 16, 17], 'cost': 450}, {'id': 9, 'elements': [1, 2, 3, 4, 5], 'cost': 215}, {'id': 10, 'elements': [1, 2, 3, 4], 'cost': 344}, {'id': 11, 'elements': [6, 8, 10], 'cost': 93}, {'id': 12, 'elements': [15, 16, 17, 18], 'cost': 324}, {'id': 13, 'elements': [9, 10], 'cost': 152}, {'id': 14, 'elements': [6, 8, 9, 10, 11], 'cost': 245}, {'id': 15, 'elements': [8, 9, 12, 13], 'cost': 252}, {'id': 16, 'elements': [9, 10, 11, 12, 13], 'cost': 190}, {'id': 17, 'elements': [1, 4, 5, 9], 'cost': 136}, {'id': 18, 'elements': [16, 17], 'cost': 36}, {'id': 19, 'elements': [12, 15], 'cost': 140}, {'id': 20, 'elements': [7, 9, 10, 11, 12], 'cost': 155}, {'id': 21, 'elements': [2, 5, 7], 'cost': 9}, {'id': 22, 'elements': [5, 6], 'cost': 68}, {'id': 23, 'elements': [3, 4, 8], 'cost': 138}, {'id': 24, 'elements': [6, 7, 8, 9], 'cost': 8}, {'id': 25, 'elements': [1, 2, 4, 5], 'cost': 156}, {'id': 26, 'elements': [2, 3, 4, 6], 'cost': 256}, {'id': 27, 'elements': [8, 9, 10, 12], 'cost': 80}, {'id': 28, 'elements': [13, 16, 18], 'cost': 186}, {'id': 29, 'elements': [1, 2, 3, 5, 6], 'cost': 415}, {'id': 30, 'elements': [15, 17, 18], 'cost': 216}, {'id': 31, 'elements': [12, 13, 15], 'cost': 255}, {'id': 32, 'elements': [11, 14, 15, 16], 'cost': 360}, {'id': 33, 'elements': [11, 12, 13, 15], 'cost': 16}, {'id': 34, 'elements': [13, 14, 16], 'cost': 216}, {'id': 35, 'elements': [11, 13, 17], 'cost': 294}, {'id': 36, 'elements': [8, 13, 15], 'cost': 222}, {'id': 37, 'elements': [4, 6, 7, 8, 10], 'cost': 280}, {'id': 38, 'elements': [7, 9], 'cost': 194}, {'id': 39, 'elements': [4, 5, 6], 'cost': 138}, {'id': 40, 'elements': [3, 7, 8, 11], 'cost': 112}, {'id': 41, 'elements': [11, 13, 16, 18], 'cost': 132}, {'id': 42, 'elements': [14, 16, 18], 'cost': 9}, {'id': 43, 'elements': [9, 10, 11, 12, 14], 'cost': 65}, {'id': 44, 'elements': [1, 3, 4], 'cost': 219}, {'id': 45, 'elements': [3, 5, 6], 'cost': 288}, {'id': 46, 'elements': [15, 16], 'cost': 138}, {'id': 47, 'elements': [1], 'cost': 10000, 'singleton': True}, {'id': 48, 'elements': [2], 'cost': 10000, 'singleton': True}, {'id': 49, 'elements': [3], 'cost': 10000, 'singleton': True}, {'id': 50, 'elements': [4], 'cost': 10000, 'singleton': True}, {'id': 51, 'elements': [5], 'cost': 10000, 'singleton': True}, {'id': 52, 'elements': [6], 'cost': 10000, 'singleton': True}, {'id': 53, 'elements': [7], 'cost': 10000, 'singleton': True}, {'id': 54, 'elements': [8], 'cost': 10000, 'singleton': True}, {'id': 55, 'elements': [9], 'cost': 10000, 'singleton': True}, {'id': 56, 'elements': [10], 'cost': 10000, 'singleton': True}, {'id': 57, 'elements': [11], 'cost': 10000, 'singleton': True}, {'id': 58, 'elements': [12], 'cost': 10000, 'singleton': True}, {'id': 59, 'elements': [13], 'cost': 10000, 'singleton': True}, {'id': 60, 'elements': [14], 'cost': 10000, 'singleton': True}, {'id': 61, 'elements': [15], 'cost': 10000, 'singleton': True}, {'id': 62, 'elements': [16], 'cost': 10000, 'singleton': True}, {'id': 63, 'elements': [17], 'cost': 10000, 'singleton': True}, {'id': 64, 'elements': [18], 'cost': 10000, 'singleton': True}], 'generator_config': {'items_min': 8, 'items_max': 25, 'bag_factor_min': 3.0, 'bag_factor_max': 5.0, 'high_cost': 10000, 'cluster_sigma_ratio': 0.1}, 'viz_matrix': 'generated_data/SPP/viz/SPP_M/instance_0049_matrix.png', 'viz_bag_stats': 'generated_data/SPP/viz/SPP_M/instance_0049_bag_stats.png'}","[1, 9, 11, 19, 38, 41]",838.0,"{'num_elements': 18, 'num_sets': 64, 'sets': [{'id': 'S1', 'elements': [14, 17], 'cost': 64}, {'id': 'S2', 'elements': [7, 11], 'cost': 74}, {'id': 'S3', 'elements': [15, 16, 18], 'cost': 156}, {'id': 'S4', 'elements': [4, 7], 'cost': 48}, {'id': 'S5', 'elements': [16, 17, 18], 'cost': 231}, {'id': 'S6', 'elements': [1, 2, 3], 'cost': 258}, {'id': 'S7', 'elements': [5, 7, 8, 9, 10], 'cost': 280}, {'id': 'S8', 'elements': [11, 13, 14, 16, 17], 'cost': 450}, {'id': 'S9', 'elements': [1, 2, 3, 4, 5], 'cost': 215}, {'id': 'S10', 'elements': [1, 2, 3, 4], 'cost': 344}, {'id': 'S11', 'elements': [6, 8, 10], 'cost': 93}, {'id': 'S12', 'elements': [15, 16, 17, 18], 'cost': 324}, {'id': 'S13', 'elements': [9, 10], 'cost': 152}, {'id': 'S14', 'elements': [6, 8, 9, 10, 11], 'cost': 245}, {'id': 'S15', 'elements': [8, 9, 12, 13], 'cost': 252}, {'id': 'S16', 'elements': [9, 10, 11, 12, 13], 'cost': 190}, {'id': 'S17', 'elements': [1, 4, 5, 9], 'cost': 136}, {'id': 'S18', 'elements': [16, 17], 'cost': 36}, {'id': 'S19', 'elements': [12, 15], 'cost': 140}, {'id': 'S20', 'elements': [7, 9, 10, 11, 12], 'cost': 155}, {'id': 'S21', 'elements': [2, 5, 7], 'cost': 9}, {'id': 'S22', 'elements': [5, 6], 'cost': 68}, {'id': 'S23', 'elements': [3, 4, 8], 'cost': 138}, {'id': 'S24', 'elements': [6, 7, 8, 9], 'cost': 8}, {'id': 'S25', 'elements': [1, 2, 4, 5], 'cost': 156}, {'id': 'S26', 'elements': [2, 3, 4, 6], 'cost': 256}, {'id': 'S27', 'elements': [8, 9, 10, 12], 'cost': 80}, {'id': 'S28', 'elements': [13, 16, 18], 'cost': 186}, {'id': 'S29', 'elements': [1, 2, 3, 5, 6], 'cost': 415}, {'id': 'S30', 'elements': [15, 17, 18], 'cost': 216}, {'id': 'S31', 'elements': [12, 13, 15], 'cost': 255}, {'id': 'S32', 'elements': [11, 14, 15, 16], 'cost': 360}, {'id': 'S33', 'elements': [11, 12, 13, 15], 'cost': 16}, {'id': 'S34', 'elements': [13, 14, 16], 'cost': 216}, {'id': 'S35', 'elements': [11, 13, 17], 'cost': 294}, {'id': 'S36', 'elements': [8, 13, 15], 'cost': 222}, {'id': 'S37', 'elements': [4, 6, 7, 8, 10], 'cost': 280}, {'id': 'S38', 'elements': [7, 9], 'cost': 194}, {'id': 'S39', 'elements': [4, 5, 6], 'cost': 138}, {'id': 'S40', 'elements': [3, 7, 8, 11], 'cost': 112}, {'id': 'S41', 'elements': [11, 13, 16, 18], 'cost': 132}, {'id': 'S42', 'elements': [14, 16, 18], 'cost': 9}, {'id': 'S43', 'elements': [9, 10, 11, 12, 14], 'cost': 65}, {'id': 'S44', 'elements': [1, 3, 4], 'cost': 219}, {'id': 'S45', 'elements': [3, 5, 6], 'cost': 288}, {'id': 'S46', 'elements': [15, 16], 'cost': 138}, {'id': 'S47', 'elements': [1], 'cost': 10000}, {'id': 'S48', 'elements': [2], 'cost': 10000}, {'id': 'S49', 'elements': [3], 'cost': 10000}, {'id': 'S50', 'elements': [4], 'cost': 10000}, {'id': 'S51', 'elements': [5], 'cost': 10000}, {'id': 'S52', 'elements': [6], 'cost': 10000}, {'id': 'S53', 'elements': [7], 'cost': 10000}, {'id': 'S54', 'elements': [8], 'cost': 10000}, {'id': 'S55', 'elements': [9], 'cost': 10000}, {'id': 'S56', 'elements': [10], 'cost': 10000}, {'id': 'S57', 'elements': [11], 'cost': 10000}, {'id': 'S58', 'elements': [12], 'cost': 10000}, {'id': 'S59', 'elements': [13], 'cost': 10000}, {'id': 'S60', 'elements': [14], 'cost': 10000}, {'id': 'S61', 'elements': [15], 'cost': 10000}, {'id': 'S62', 'elements': [16], 'cost': 10000}, {'id': 'S63', 'elements': [17], 'cost': 10000}, {'id': 'S64', 'elements': [18], 'cost': 10000}]}","['S1', 'S9', 'S11', 'S19', 'S38', 'S41']",50,markdown_table,1