| import OAI.Combinatorics.MatchingPSD.MatchingConversion |
|
|
| namespace OAI |
|
|
| namespace PerfectMatchingPSD |
| open MatchingSmoothing Matrix |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma perfectMatching_nonempty {n : ℕ} (hn : Even n) : Nonempty (PerfectMatching n) := by |
| classical |
| obtain ⟨k,hk⟩ := hn |
| let e : (Fin k × Bool) ≃ Fin n := Fintype.equivOfCardEq (by |
| simp only [Fintype.card_prod, Fintype.card_fin, Fintype.card_bool] |
| omega) |
| exact ⟨fromPairing (Pairing.transport e standardPairing)⟩ |
|
|
| lemma slack_le_vertices {n : ℕ} (hn : 1 ≤ n) (i : Row n) (M : PerfectMatching n) : |
| slack i M ≤ (n:ℝ) := by |
| rcases i with e | U |
| · simp only [slack] |
| split_ifs |
| · exact_mod_cast hn |
| · positivity |
| · have hd := cut_degree_identity U.1 M |
| have hcard : U.1.card ≤ n := by simpa using Finset.card_le_univ U.1 |
| have hc : crossingCount U.1 M ≤ n := by omega |
| have hc' : (crossingCount U.1 M : ℝ) ≤ n := by exact_mod_cast hc |
| change (crossingCount U.1 M : ℝ)-1 ≤ (n:ℝ) |
| linarith |
|
|
| lemma normalized_factorization {n r : ℕ} (hn : Even n) (hnpos : 1 ≤ n) |
| (hfac : HasFactorization n r) : |
| ∃ (A : Row n → Matrix (Fin r) (Fin r) ℝ) |
| (B : PerfectMatching n → Matrix (Fin r) (Fin r) ℝ), |
| (∀ i, (A i).PosSemidef) ∧ (∀ M, (B M).PosSemidef) ∧ |
| (∀ i M, (A i*B M).trace = slack i M) ∧ |
| (∀ i, (A i).trace ≤ (n:ℝ)+1) ∧ (∀ M, (B M).trace ≤ (r:ℝ)) := by |
| classical |
| have := perfectMatching_nonempty hn |
| rcases hfac with ⟨A,B,hA,hB,hpair⟩ |
| obtain ⟨A',B',hA',hB',hpair',htrA,htrB⟩ := exists_normalized_factors A B hA hB (n:ℝ) |
| (fun i M => by rw [← hpair]; exact slack_le_vertices hnpos i M) |
| exact ⟨A',B',hA',hB',fun i M => (hpair' i M).trans (hpair i M).symm, |
| htrA,by simpa using htrB⟩ |
|
|
| end PerfectMatchingPSD |
|
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| end OAI |
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|
