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e5a97c6 | 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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 | import OAI.Combinatorics.PerfectMatching.CountCorrect
namespace OAI
section
namespace BinaryMatching.Algorithm
noncomputable section
open Finset MatchingEntropy
lemma count_zero_vertices (G : Input) (hn : G.n=0) : count G=1 := by
classical
let M : (completeGraph G.n).Matching := ⟨∅,fun v => by have := v.isLt; omega⟩
have hM (N : (completeGraph G.n).Matching) : N=M := by
apply Subtype.ext
apply Finset.eq_empty_of_forall_notMem
intro e _
have := e.val.1.isLt
omega
unfold count
rw [sum_eq_single M]
· simp [M]
· intro N _ hne
exact (hne (hM N)).elim
· simp
lemma count_odd_vertices (G : Input) (hn : Odd G.n) : count G=0 := by
classical
have hempty : IsEmpty (completeGraph G.n).Matching := ⟨fun M => by
have hh := (completeGraph G.n).matching_card_twice M
simp only [Fintype.card_fin] at hh
obtain ⟨r,hr⟩ := hn
omega⟩
simp [count]
lemma count_two_vertices (G : Input) (hn : G.n=2) : count G=rawMultiplicity G 0 1 := by
classical
let e : Pair G.n := ⟨(⟨0,by omega⟩,⟨1,by omega⟩),by simp⟩
have heq (f : Pair G.n) : f=e := by
apply Subtype.ext
apply Prod.ext <;> apply Fin.ext
· have hh := f.property; have h1 := f.val.1.isLt; have h2 := f.val.2.isLt
dsimp [e] at hh ⊢; omega
· have hh := f.property; have h1 := f.val.1.isLt; have h2 := f.val.2.isLt
dsimp [e] at hh ⊢; omega
have heinc (v : Fin G.n) : (completeGraph G.n).Incident v e := by
have hv := v.isLt
change e.val.1=v ∨ e.val.2=v
by_cases hz : v.val=0
· left; apply Fin.ext; exact hz.symm
· right; apply Fin.ext; dsimp [e]; omega
let M : (completeGraph G.n).Matching := ⟨{e},fun v => ⟨e,⟨by simp,heinc v⟩,
fun _ h => by simpa only [mem_singleton] using h.1⟩⟩
have hM (N : (completeGraph G.n).Matching) : N=M := by
apply Subtype.ext
apply Finset.ext
intro f
have hfn := heq f
obtain ⟨g,hg,_⟩ := N.property e.val.1
have hgn := heq g
have hmem : e∈N.val := hgn ▸ hg.1
simp only [M,mem_singleton,hfn]
exact iff_true_intro hmem
unfold count
rw [sum_eq_single M]
· simp only [M,prod_singleton]
unfold multiplicity rawMultiplicity
dsimp [e]
cases G.records.find? (fun r => r.left=0 && r.right=1) <;> rfl
· intro N _ hne
exact (hne (hM N)).elim
· simp
lemma present_endpoints_covered (G : Input) (e : Pair G.n) (he : 0 < multiplicity G e) :
e.val.1.val∈covered G ∧ e.val.2.val∈covered G := by
unfold multiplicity at he
cases hf : G.records.find? (fun r => r.left=e.val.1.val && r.right=e.val.2.val) with
| none => simp only [hf] at he; omega
| some r =>
have hr := List.mem_of_find?_eq_some hf
have hp := List.find?_some hf
simp only [Bool.and_eq_true,decide_eq_true_eq] at hp
simp only [hf] at he
have hr' : r∈G.records.filter (fun r => 0 < r.multiplicity) := by simp [hr,he]
constructor
· simp only [covered,List.mem_toFinset,List.mem_flatMap]
refine ⟨r,hr',?_⟩
simp [hp.1]
· simp only [covered,List.mem_toFinset,List.mem_flatMap]
refine ⟨r,hr',?_⟩
simp [hp.2]
lemma positive_count_covered (G : Input) (hN : 0 < count G) : G.n ≤ (covered G).card := by
obtain ⟨M,hM⟩ := positive_count_valid G hN
have hsub : range G.n ⊆ covered G := by
intro v hv
let w : Fin G.n := ⟨v,mem_range.mp hv⟩
obtain ⟨e,⟨he,hinc⟩,_⟩ := M.property w
have hep := present_endpoints_covered G e (hM e he)
rcases hinc with h|h
· have hh : e.val.1.val=v := congrArg Fin.val h
simpa only [hh] using hep.1
· have hh : e.val.2.val=v := congrArg Fin.val h
simpa only [hh] using hep.2
simpa only [card_range] using card_le_card hsub
lemma count_zero_of_uncovered (G : Input) (hu : (covered G).card < G.n) : count G=0 := by
by_contra hN
have := positive_count_covered G (Nat.pos_of_ne_zero hN)
omega
theorem approximate_correct (G : Input) :
approximate G ≤ count G ∧ count G ≤ 2^(9*G.n)*approximate G ∧
(approximate G=0 ↔ count G=0) := by
unfold approximate
split_ifs with hn0 hodd hn2 hcover
· rw [count_zero_vertices G hn0,hn0]
norm_num
· rw [count_odd_vertices G hodd]
simp
· rw [count_two_vertices G hn2]
refine ⟨le_rfl,?_,Iff.rfl⟩
exact Nat.le_mul_of_pos_left _ (by positivity)
· rw [count_zero_of_uncovered G hcover]
simp
· have heven : Even G.n := Nat.not_odd_iff_even.mp hodd
have hn : 4 ≤ G.n := by
obtain ⟨r,hr⟩ := heven
omega
exact estimate_correct G hn heven
end
end BinaryMatching.Algorithm
end
end OAI
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