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ea76aac | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 | import OAI.Computability.MatchingCount.OddCutUncross
namespace OAI
section
namespace BinaryMatching.OddMincut
open Finset UnitFlow UnitMincut TerminalMincut CutAlgebra OddCutParity OddCutUncross
variable {n : ℕ}
def outside (A : Finset (Fin n)) : Option (Fin n) :=
(List.finRange n).find? (fun v => decide (v∉A))
lemma outside_some (A : Finset (Fin n)) (hA : Aᶜ.Nonempty) :
∃ r, outside A=some r ∧ r∉A := by
obtain ⟨v,hv⟩ := hA
have hv' : v∉A := mem_compl.mp hv
cases he : outside A with
| none =>
have hh := List.find?_eq_none.mp he v (by simp)
simp [hv'] at hh
| some r =>
have hh := List.find?_some (show (List.finRange n).find?
(fun v => decide (v∉A))=some r from he)
exact ⟨r,rfl,by simpa using hh⟩
def cheaper (c : Fin n → Fin n → ℕ) (S T : Finset (Fin n)) : Finset (Fin n) :=
if cutCapacity c S ≤ cutCapacity c T then S else T
lemma cheaper_le_left (c : Fin n → Fin n → ℕ) (S T : Finset (Fin n)) :
cutCapacity c (cheaper c S T) ≤ cutCapacity c S := by
unfold cheaper; split_ifs <;> omega
lemma cheaper_le_right (c : Fin n → Fin n → ℕ) (S T : Finset (Fin n)) :
cutCapacity c (cheaper c S T) ≤ cutCapacity c T := by
unfold cheaper; split_ifs <;> omega
lemma cheaper_odd (c : Fin n → Fin n → ℕ) (U S T : Finset (Fin n))
(hS : OddSide U S) (hT : OddSide U T) : OddSide U (cheaper c S T) := by
unfold cheaper; split_ifs <;> assumption
def recurse : ℕ → (Fin n → Fin n → ℕ) → Finset (Fin n) → Finset (Fin n)
| 0,_,_ => ∅
| k+1,c,U =>
let A := terminalCut c U
if OddSide U A then A else
match outside A, outside Aᶜ with
| some r, some q =>
cheaper c
(lift (contractMap A r) (recurse k (push c (contractMap A r)) (A∩U)))
(lift (contractMap Aᶜ q) (recurse k (push c (contractMap Aᶜ q)) (Aᶜ∩U)))
| _,_ => ∅
lemma recurse_correct (fuel : ℕ) (c : Fin n → Fin n → ℕ)
(hc : ∀ i j, c i j=c j i) (U : Finset (Fin n)) (he : Even U.card)
(hU : 2≤U.card) (hbound : U.card≤fuel) :
OddSide U (recurse fuel c U) ∧
∀ S, OddSide U S → cutCapacity c (recurse fuel c U) ≤ cutCapacity c S := by
induction fuel generalizing c U with
| zero => omega
| succ k ih =>
let A := terminalCut c U
obtain ⟨hsep,hmin⟩ := terminalCut_correct c U hU
change Separates U A at hsep
change ∀ T, Separates U T → cutCapacity c A ≤ cutCapacity c T at hmin
by_cases hodd : OddSide U A
· have heq : recurse (k+1) c U=A := by
rw [recurse]
change (if OddSide U A then A else _)=A
exact ite_eq_left hodd
rw [heq]
exact ⟨hodd,fun S hS => hmin S (odd_separates U S he hS)⟩
· have heA : Even (A∩U).card := by
simp only [OddSide,Nat.odd_iff] at hodd
rw [Nat.even_iff]; omega
have heB := even_other U A he heA
obtain ⟨hposA,hposB⟩ := separate_cards U A hsep
have hpart := terminal_partition U A
have htwoA : 2≤(A∩U).card := by rw [Nat.even_iff] at heA; omega
have htwoB : 2≤(Aᶜ∩U).card := by rw [Nat.even_iff] at heB; omega
have hboundA : (A∩U).card≤k := by omega
have hboundB : (Aᶜ∩U).card≤k := by omega
have hnonA : A.Nonempty := by
obtain ⟨v,hv⟩ := card_pos.mp hposA
exact ⟨v,(mem_inter.mp hv).1⟩
have hnonB : Aᶜ.Nonempty := by
obtain ⟨v,hv⟩ := card_pos.mp hposB
exact ⟨v,(mem_inter.mp hv).1⟩
obtain ⟨r,hr,hrA⟩ := outside_some A hnonB
obtain ⟨q,hq,hqB⟩ := outside_some Aᶜ (by simpa using hnonA)
let f := contractMap A r
let g := contractMap Aᶜ q
let X := recurse k (push c f) (A∩U)
let Y := recurse k (push c g) (Aᶜ∩U)
obtain ⟨hX,minX⟩ := ih (push c f) (push_symm c hc f) (A∩U) heA htwoA hboundA
obtain ⟨hY,minY⟩ := ih (push c g) (push_symm c hc g) (Aᶜ∩U) heB htwoB hboundB
change OddSide (A∩U) X at hX
change OddSide (Aᶜ∩U) Y at hY
have hLX : OddSide U (lift f X) := (odd_lift_iff U A X r heB).2 hX
have hLY : OddSide U (lift g Y) :=
(odd_lift_iff U Aᶜ Y q (by simpa using heA)).2 hY
have heq : recurse (k+1) c U=cheaper c (lift f X) (lift g Y) := by
rw [recurse]
change (if OddSide U A then A else _) = _
rw [ite_eq_right hodd,hr,hq]
rw [heq]
refine ⟨cheaper_odd c U _ _ hLX hLY,?_⟩
intro S hS
by_cases hSA : OddSide (A∩U) S
· obtain ⟨T,hTA,hT,hcap⟩ := uncross_on_side c hc U A S hsep heA hmin hSA
have hchild := (odd_subset_iff U A T hTA).2 hT
have hminT := minX T hchild
change cutCapacity (push c f) X ≤ cutCapacity (push c f) T at hminT
rw [cut_push] at hminT
rw [show f=contractMap A r from rfl,cut_contract_subset c A T r hrA hTA] at hminT
exact (cheaper_le_left c _ _).trans (hminT.trans hcap)
· have hSB := split_odd U S A hS hSA
have hminB : ∀ T, Separates U T → cutCapacity c Aᶜ ≤ cutCapacity c T := by
simpa only [cut_compl c hc] using hmin
obtain ⟨T,hTB,hT,hcap⟩ := uncross_on_side c hc U Aᶜ S
(separates_compl U A hsep) heB hminB hSB
have hchild := (odd_subset_iff U Aᶜ T hTB).2 hT
have hminT := minY T hchild
change cutCapacity (push c g) Y ≤ cutCapacity (push c g) T at hminT
rw [cut_push] at hminT
rw [show g=contractMap Aᶜ q from rfl,cut_contract_subset c Aᶜ T q hqB hTB] at hminT
exact (cheaper_le_right c _ _).trans (hminT.trans hcap)
def oddCut (c : Fin n → Fin n → ℕ) (U : Finset (Fin n)) : Finset (Fin n) :=
recurse U.card c U
theorem oddCut_correct (c : Fin n → Fin n → ℕ) (hc : ∀ i j, c i j=c j i)
(U : Finset (Fin n)) (he : Even U.card) (hU : 2≤U.card) :
OddSide U (oddCut c U) ∧
∀ S, OddSide U S → cutCapacity c (oddCut c U) ≤ cutCapacity c S :=
recurse_correct U.card c hc U he hU le_rfl
end BinaryMatching.OddMincut
end
end OAI
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