File size: 2,329 Bytes
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 | 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
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