| import OAI.Computability.LoopMatching.ExpandBijection |
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| namespace OAI |
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| /-! Polynomial-time approximation of weighted matchings with singleton loops. -/ |
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|
| noncomputable section |
| open scoped BigOperators |
| namespace LoopMatching |
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| lemma expanded_weight_zero_of_not_supported (G : Input) (j : ℕ) |
| (N : (BinaryMatching.completeGraph (expand G j).n).Matching) |
| (h : ¬ ∀ e ∈ N.val, e.val.1.val < G.n) : |
| (∏ e ∈ N.val, BinaryMatching.multiplicity (expand G j) e) = 0 := by |
| classical |
| push Not at h |
| obtain ⟨e, he, hl⟩ := h |
| have heleft := e.val.1.isLt |
| have heright := e.val.2.isLt |
| have hord := e.property |
| change e.val.1.val < e.val.2.val at hord |
| simp only [expand_n] at heleft heright |
| let f : BinaryMatching.Pair j := |
| ⟨(⟨e.val.1.val - G.n, by omega⟩, |
| ⟨e.val.2.val - G.n, by omega⟩), by |
| change e.val.1.val - G.n < e.val.2.val - G.n |
| omega⟩ |
| have hf : dummyPair G j f = e := by |
| apply Subtype.ext |
| apply Prod.ext <;> apply Fin.ext |
| · dsimp [dummyPair, f]; omega |
| · dsimp [dummyPair, f]; omega |
| apply Finset.prod_eq_zero he |
| rw [← hf] |
| exact multiplicity_expand_dummy G j f |
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|
| theorem expanded_count (G : Input) (j : ℕ) : |
| BinaryMatching.count (expand G j) = j.factorial * countWithLoops G j := by |
| classical |
| let w : (BinaryMatching.completeGraph (expand G j).n).Matching → ℕ := |
| fun N => ∏ e ∈ N.val, BinaryMatching.multiplicity (expand G j) e |
| have hzero : |
| (∑ N : {N : (BinaryMatching.completeGraph (expand G j).n).Matching // |
| ¬ ∀ e ∈ N.val, e.val.1.val < G.n}, w N.val) = 0 := by |
| apply Finset.sum_eq_zero |
| intro N _ |
| exact expanded_weight_zero_of_not_supported G j N.val N.property |
| have hs : BinaryMatching.count (expand G j) = |
| ∑ N : SupportedMatching G j, w N.val := by |
| have ht := Fintype.sum_subtype_add_sum_subtype |
| (fun N : (BinaryMatching.completeGraph (expand G j).n).Matching => |
| ∀ e ∈ N.val, e.val.1.val < G.n) w |
| rw [hzero, Nat.add_zero] at ht |
| have hi : |
| (Subtype.fintype (fun N : |
| (BinaryMatching.completeGraph (expand G j).n).Matching => |
| ∀ e ∈ N.val, e.val.1.val < G.n) : Fintype (SupportedMatching G j)) = |
| supportedMatchingFintype G j := Subsingleton.elim _ _ |
| rw [hi] at ht |
| exact ht.symm |
| rw [hs] |
| calc |
| _ = ∑ q : (M : PartialMatching G.n) × (Fin j ≃ ↥(loopVertices M)), |
| weight G q.1 := by |
| apply Fintype.sum_equiv (supportedMatchingEquiv G j) |
| intro N |
| rw [supportedMatchingEquiv_fst] |
| exact supported_matching_weight G j N |
| _ = ∑ M : PartialMatching G.n, |
| Fintype.card (Fin j ≃ ↥(loopVertices M)) * weight G M := by |
| rw [Fintype.sum_sigma] |
| apply Finset.sum_congr rfl |
| intro M _ |
| simp |
| _ = j.factorial * countWithLoops G j := by |
| unfold countWithLoops |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro M _ |
| rw [card_dummyAssignments] |
| by_cases h : (loopVertices M).card = j <;> simp [h] |
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| end LoopMatching |
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| end |
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| end OAI |
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