| import OAI.Computability.BinPacking.Inventory.EdgeLongConsumption |
| import OAI.Computability.BinPacking.Packing.ExtractedCover |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace BinPackingGap |
|
|
| private def leftIncidencePosition (G : GraphInput) (R : ℕ) (edge : G.Edge) (j : Fin R) : |
| G.IncidencePosition R (G.left edge) := |
| (⟨(edge, 0), G.endpoint_zero edge⟩, j) |
|
|
| private def rightIncidencePosition (G : GraphInput) (R : ℕ) (edge : G.Edge) (j : Fin R) : |
| G.IncidencePosition R (G.right edge) := |
| (⟨(edge, 1), G.endpoint_one edge⟩, j) |
|
|
| theorem extractedCover_isCover (c : ℕ) (ρ : ℝ) (G : GraphInput) (k : ℕ) |
| (hk : k ≤ G.n) {b : ℕ} (p : Packing (reductionInstance c ρ G k) b) |
| (hb : b ≤ reductionBinBound c ρ G k + c) : |
| G.IsCover (extractedCover c ρ G k p) := by |
| classical |
| let D := reductionInventory c ρ G k |
| let Sleft : G.Edge → Finset (Fin D.R) := fun edge => |
| PackingCounts.goodRepetitions p D.itemEquiv (G.left edge) edge |
| let Sright : G.Edge → Finset (Fin D.R) := fun edge => |
| PackingCounts.goodRepetitions p D.itemEquiv (G.right edge) edge |
| let Lleft : G.Edge → Fin D.R → ℕ := fun edge j => |
| (D.edgeLongTuplesAt p edge (0, j)).card |
| let Lright : G.Edge → Fin D.R → ℕ := fun edge j => |
| (D.edgeLongTuplesAt p edge (1, j)).card |
| have hL : 1 ≤ D.L := Nat.succ_le_of_lt (reductionInventory_height_pos c ρ G k) |
| apply thresholdCover_isCover G (PackingCounts.xplus p D.itemEquiv) |
| Sleft Sright Lleft Lright (depthDemand_one_le c ρ) (depthDemand_ge_72 c ρ) |
| (reductionInventory_R c ρ G k) |
| · intro edge |
| exact PackingCounts.goodRepetitions_card_ge_real p D.itemEquiv hL (G.left edge) edge |
| · intro edge |
| exact PackingCounts.goodRepetitions_card_ge_real p D.itemEquiv hL (G.right edge) edge |
| · intro edge j hj |
| have hj' : (edge, j) ∉ PackingCounts.excludedAt p D.itemEquiv (G.left edge) := |
| (mem_nonexcludedRepetitions _ _).mp hj |
| exact D.long_matches_at_nonexcluded (reductionInventory_plus_height c ρ G k) |
| (reductionInventory_minus_height c ρ G k) hL hk p hb (G.left edge) |
| (leftIncidencePosition G D.R edge j) hj' |
| · intro edge j hj |
| have hj' : (edge, j) ∉ PackingCounts.excludedAt p D.itemEquiv (G.right edge) := |
| (mem_nonexcludedRepetitions _ _).mp hj |
| exact D.long_matches_at_nonexcluded (reductionInventory_plus_height c ρ G k) |
| (reductionInventory_minus_height c ρ G k) hL hk p hb (G.right edge) |
| (rightIncidencePosition G D.R edge j) hj' |
| · intro edge |
| have hcap := D.edge_long_matches_capacity hk p hb edge Finset.univ |
| have htwo : (Finset.univ : Finset (Fin 2)) = {0, 1} := by decide |
| change (∑ j : Fin D.R, Lleft edge j) + (∑ j : Fin D.R, Lright edge j) ≤ |
| D.d * D.R + 3 * D.graph.n * lossAllowance c |
| simpa [htwo, Lleft, Lright] using hcap |
|
|
| theorem packing_extracts_cover (c : ℕ) (ρ : ℝ) (hρ : 0 < ρ) |
| (G : GraphInput) (k : ℕ) (hk : k ≤ G.n) {b : ℕ} |
| (p : Packing (reductionInstance c ρ G k) b) |
| (hb : b ≤ reductionBinBound c ρ G k + c) : |
| ∃ C : Finset G.Vertex, G.IsCover C ∧ (C.card : ℝ) ≤ (k : ℝ) + ρ * G.n := |
| ⟨extractedCover c ρ G k p, extractedCover_isCover c ρ G k hk p hb, |
| extractedCover_card_le c ρ hρ G k hk p hb⟩ |
|
|
| end BinPackingGap |
|
|
| end |
|
|
| namespace BinPackingGap |
|
|
| theorem completeFourGraph_reduction_no_packing (c : ℕ) : |
| ¬ HasPacking (reductionInstance c (1 / 8) completeFourGraph 2) |
| (reductionBinBound c (1 / 8) completeFourGraph 2 + c) := by |
| rintro ⟨p⟩ |
| exact completeFourGraph_no_small_cover |
| (packing_extracts_cover c (1 / 8) (by norm_num) completeFourGraph 2 |
| (by decide) p le_rfl) |
|
|
| theorem completeFourGraph_reduction_opt_gap (c : ℕ) : |
| reductionBinBound c (1 / 8) completeFourGraph 2 + c < |
| opt (reductionInstance c (1 / 8) completeFourGraph 2) := |
| (not_hasPacking_iff_lt_opt _ _).mp (completeFourGraph_reduction_no_packing c) |
|
|
| end BinPackingGap |
|
|
| end OAI |
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|
