| import OAI.Computability.MatchingCount.OddCutParity |
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| namespace OAI |
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| section |
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| namespace BinaryMatching.OddCutUncross |
| open Finset UnitFlow CutAlgebra TerminalMincut OddCutParity |
| variable {n : ℕ} |
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| lemma separates_compl (U A : Finset (Fin n)) (hA : Separates U A) : Separates U Aᶜ := by |
| obtain ⟨a,ha,z,hz,haa,hza⟩ := hA |
| exact ⟨z,hz,a,ha,mem_compl.mpr hza,by simpa using haa⟩ |
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| lemma uncross_on_side (c : Fin n → Fin n → ℕ) (hc : ∀ i j, c i j=c j i) |
| (U A S : Finset (Fin n)) (hsep : Separates U A) (heven : Even (A∩U).card) |
| (hmin : ∀ T, Separates U T → cutCapacity c A ≤ cutCapacity c T) |
| (hinside : OddSide (A∩U) S) : |
| ∃ T, T⊆A ∧ OddSide U T ∧ cutCapacity c T ≤ cutCapacity c S := by |
| obtain ⟨a,ha,b,hb,haa,hba⟩ := hsep |
| have horient : ∃ S', OddSide (A∩U) S' ∧ cutCapacity c S'=cutCapacity c S ∧ |
| ∃ z∈U, z∉A ∧ z∉S' := by |
| by_cases hex : ∃ z∈U, z∉A ∧ z∉S |
| · exact ⟨S,hinside,rfl,hex⟩ |
| · have hbS : b∈S := by |
| by_contra hh |
| exact hex ⟨b,hb,hba,hh⟩ |
| exact ⟨Sᶜ,odd_other (A∩U) S heven hinside,cut_compl c hc S, |
| b,hb,hba,by simpa using hbS⟩ |
| obtain ⟨S',hodd,hcap,z,hz,hza,hzs⟩ := horient |
| have hunion : Separates U (S'∪A) := |
| ⟨a,ha,z,hz,mem_union_right _ haa,by simpa only [mem_union,not_or] using And.intro hzs hza⟩ |
| have hlow := hmin (S'∪A) hunion |
| have hsub := cut_submodular c S' A |
| refine ⟨S'∩A,inter_subset_right,?_,?_⟩ |
| · simpa only [OddSide,inter_assoc] using hodd |
| · omega |
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| end BinaryMatching.OddCutUncross |
| end |
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| end OAI |
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