File size: 2,736 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 79 80 81 82 | import OAI.Combinatorics.PerfectMatching.Model
namespace OAI
/-! Polynomial-time approximation of weighted matchings with singleton loops. -/
open scoped BigOperators
namespace LoopMatching
structure LoopRecord where
vertex : ℕ
multiplicity : ℕ
deriving DecidableEq
structure Input where
ordinary : BinaryMatching.Input
loops : List LoopRecord
valid : ∀ e ∈ loops, e.vertex < ordinary.n
unique : (loops.map LoopRecord.vertex).Nodup
abbrev Input.n (G : Input) : ℕ := G.ordinary.n
def loopMultiplicity (G : Input) (v : Fin G.n) : ℕ :=
match G.loops.find? (fun r => r.vertex == v.val) with
| none => 0
| some r => r.multiplicity
def IsPartialMatching (n : ℕ) (M : Finset (BinaryMatching.Pair n)) : Prop :=
∀ v : Fin n, ∀ e ∈ M, ∀ f ∈ M,
(BinaryMatching.completeGraph n).Incident v e →
(BinaryMatching.completeGraph n).Incident v f → e = f
abbrev PartialMatching (n : ℕ) :=
{ M : Finset (BinaryMatching.Pair n) // IsPartialMatching n M }
noncomputable instance partialMatchingFintype (n : ℕ) : Fintype (PartialMatching n) :=
Fintype.ofFinite _
noncomputable def loopVertices {n : ℕ} (M : PartialMatching n) : Finset (Fin n) := by
classical
exact Finset.univ.filter (fun v =>
∀ e ∈ M.val, ¬ (BinaryMatching.completeGraph n).Incident v e)
noncomputable def weight (G : Input) (M : PartialMatching G.n) : ℕ :=
(∏ e ∈ M.val, BinaryMatching.multiplicity G.ordinary e) *
∏ v ∈ loopVertices M, loopMultiplicity G v
noncomputable def count (G : Input) : ℕ := ∑ M : PartialMatching G.n, weight G M
noncomputable def countWithLoops (G : Input) (j : ℕ) : ℕ := by
classical
exact ∑ M : PartialMatching G.n,
if (loopVertices M).card = j then weight G M else 0
def encodeLoopRecord (r : LoopRecord) : List Bool :=
BinaryMatching.encodeNat r.vertex ++ BinaryMatching.encodeNat r.multiplicity
def encodeInput (G : Input) : List Bool :=
BinaryMatching.encodeInput G.ordinary ++
BinaryMatching.encodeNat G.loops.length ++ G.loops.flatMap encodeLoopRecord
def encodeOutput (a : ℕ × ℕ) : List Bool :=
BinaryMatching.encodeNat a.1 ++ BinaryMatching.encodeNat a.2
def outputValue (a : ℕ × ℕ) : ℚ := (a.1 : ℚ) / a.2
def FullEndpoint : Prop :=
∃ A : Input → ℕ × ℕ,
∃ machine : Turing.TM2ComputableInPolyTime encodeInput encodeOutput A,
(∀ k, Finite (machine.tm.Γ k)) ∧
(∃ P : Polynomial ℕ, ∀ G,
(encodeOutput (A G)).length ≤ P.eval (encodeInput G).length) ∧
∀ G, 0 < (A G).2 ∧
(A G).1 ≤ (A G).2 * count G ∧
(A G).2 * count G ≤ 2 ^ (18 * G.n) * (A G).1 ∧
((A G).1 = 0 ↔ count G = 0)
end LoopMatching
end OAI
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