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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 | import OAI.Computability.LoopMatching.ExpandBijection
namespace OAI
/-! Polynomial-time approximation of weighted matchings with singleton loops. -/
noncomputable section
open scoped BigOperators
namespace LoopMatching
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
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]
end LoopMatching
end
end OAI
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