| import OAI.Computability.LoopMatching.Model |
| import Mathlib.Data.Fintype.BigOperators |
|
|
| namespace OAI |
|
|
| /-! Polynomial-time approximation of weighted matchings with singleton loops. -/ |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace LoopMatching |
|
|
| def OrdinaryChoices (G : Input) (M : PartialMatching G.n) : Type := |
| (e : M.val) → Fin (BinaryMatching.multiplicity G.ordinary e.val) |
|
|
| def LoopChoices (G : Input) (M : PartialMatching G.n) : Type := |
| (v : loopVertices M) → Fin (loopMultiplicity G v.val) |
|
|
| def MatchingChoices (G : Input) (M : PartialMatching G.n) : Type := |
| OrdinaryChoices G M × LoopChoices G M |
|
|
| def Choices (G : Input) : Type := |
| (M : PartialMatching G.n) × MatchingChoices G M |
| |
| instance ordinaryChoicesFintype (G : Input) (M : PartialMatching G.n) : |
| Fintype (OrdinaryChoices G M) := by |
| classical |
| unfold OrdinaryChoices |
| infer_instance |
| |
| instance loopChoicesFintype (G : Input) (M : PartialMatching G.n) : |
| Fintype (LoopChoices G M) := by |
| classical |
| unfold LoopChoices |
| infer_instance |
| |
| instance matchingChoicesFintype (G : Input) (M : PartialMatching G.n) : |
| Fintype (MatchingChoices G M) := by |
| unfold MatchingChoices |
| infer_instance |
| |
| instance choicesFintype (G : Input) : Fintype (Choices G) := by |
| unfold Choices |
| infer_instance |
| |
| theorem card_ordinaryChoices (G : Input) (M : PartialMatching G.n) : |
| Fintype.card (OrdinaryChoices G M) = |
| ∏ e ∈ M.val, BinaryMatching.multiplicity G.ordinary e := by |
| classical |
| unfold OrdinaryChoices |
| simp only [Fintype.card_pi, Fintype.card_fin, Finset.prod_coe_sort] |
| |
| theorem card_loopChoices (G : Input) (M : PartialMatching G.n) : |
| Fintype.card (LoopChoices G M) = |
| ∏ v ∈ loopVertices M, loopMultiplicity G v := by |
| classical |
| unfold LoopChoices |
| simp only [Fintype.card_pi, Fintype.card_fin, Finset.prod_coe_sort] |
| |
| theorem card_matchingChoices (G : Input) (M : PartialMatching G.n) : |
| Fintype.card (MatchingChoices G M) = weight G M := by |
| unfold MatchingChoices |
| rw [Fintype.card_prod, card_ordinaryChoices, card_loopChoices] |
| rfl |
| |
| theorem count_eq_card_choices (G : Input) : |
| count G = Fintype.card (Choices G) := by |
| classical |
| unfold Choices |
| rw [Fintype.card_sigma] |
| unfold count |
| exact Finset.sum_congr rfl (fun M _ => (card_matchingChoices G M).symm) |
| |
| end LoopMatching |
| |
| end |
| |
| end OAI |
| |
