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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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | import OAI.Computability.LoopMatching.Model
import OAI.Computability.LoopMatching.CountDecomposition
namespace OAI
/-! Polynomial-time approximation of weighted matchings with singleton loops. -/
namespace LoopMatching
def recordedVertices (G : Input) : Finset ℕ :=
(G.ordinary.records.map BinaryMatching.Record.left ++
G.ordinary.records.map BinaryMatching.Record.right ++
G.loops.map LoopRecord.vertex).toFinset
theorem recordedVertices_card_le (G : Input) :
(recordedVertices G).card ≤ 2 * G.ordinary.records.length + G.loops.length := by
have h := List.toFinset_card_le (G.ordinary.records.map BinaryMatching.Record.left ++
G.ordinary.records.map BinaryMatching.Record.right ++
G.loops.map LoopRecord.vertex)
simpa only [recordedVertices, List.length_append, List.length_map, two_mul] using h
theorem ordinary_endpoints_mem_recordedVertices (G : Input)
(e : BinaryMatching.Pair G.n) (he : BinaryMatching.multiplicity G.ordinary e ≠ 0) :
e.val.1.val ∈ recordedVertices G ∧ e.val.2.val ∈ recordedVertices G := by
unfold BinaryMatching.multiplicity at he
cases hf : G.ordinary.records.find?
(fun r => r.left = e.val.1.val && r.right = e.val.2.val) with
| none => simp only [hf] at he; exact (he rfl).elim
| 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
constructor
· have hm : e.val.1.val ∈ G.ordinary.records.map BinaryMatching.Record.left :=
List.mem_map.mpr ⟨r, hr, hp.1⟩
simp only [recordedVertices, List.mem_toFinset, List.mem_append]
exact Or.inl (Or.inl hm)
· have hm : e.val.2.val ∈ G.ordinary.records.map BinaryMatching.Record.right :=
List.mem_map.mpr ⟨r, hr, hp.2⟩
simp only [recordedVertices, List.mem_toFinset, List.mem_append]
exact Or.inl (Or.inr hm)
theorem loop_mem_recordedVertices (G : Input) (v : Fin G.n)
(hv : loopMultiplicity G v ≠ 0) : v.val ∈ recordedVertices G := by
unfold loopMultiplicity at hv
cases hf : G.loops.find? (fun r => r.vertex == v.val) with
| none => simp only [hf] at hv; exact (hv rfl).elim
| some r =>
have hr := List.mem_of_find?_eq_some hf
have hp := List.find?_some hf
have heq : r.vertex = v.val := by simpa using hp
have hm : v.val ∈ G.loops.map LoopRecord.vertex := List.mem_map.mpr ⟨r, hr, heq⟩
simp only [recordedVertices, List.mem_toFinset, List.mem_append]
exact Or.inr hm
theorem range_subset_recordedVertices_of_weight_ne_zero (G : Input)
(M : PartialMatching G.n) (hM : weight G M ≠ 0) :
Finset.range G.n ⊆ recordedVertices G := by
classical
intro x hx
let v : Fin G.n := ⟨x, Finset.mem_range.mp hx⟩
by_cases hloop : v ∈ loopVertices M
· have hn : loopMultiplicity G v ≠ 0 := by
intro hz
exact hM (weight_eq_zero_of_loopMultiplicity_eq_zero G M v hloop hz)
exact loop_mem_recordedVertices G v hn
· have hex : ∃ e ∈ M.val, (BinaryMatching.completeGraph G.n).Incident v e := by
simpa only [loopVertices, Finset.mem_filter, Finset.mem_univ, true_and,
not_forall, not_not, exists_prop] using hloop
obtain ⟨e, he, hi⟩ := hex
have hn : BinaryMatching.multiplicity G.ordinary e ≠ 0 := by
intro hz
exact hM (weight_eq_zero_of_multiplicity_eq_zero G M e he hz)
have hep := ordinary_endpoints_mem_recordedVertices G e hn
change e.val.1 = v ∨ e.val.2 = v at hi
rcases hi with hi | hi
· simpa only [hi] using hep.1
· simpa only [hi] using hep.2
theorem vertex_bound_of_weight_ne_zero (G : Input) (M : PartialMatching G.n)
(hM : weight G M ≠ 0) :
G.n ≤ 2 * G.ordinary.records.length + G.loops.length := by
have h := Finset.card_le_card (range_subset_recordedVertices_of_weight_ne_zero G M hM)
rw [Finset.card_range] at h
exact h.trans (recordedVertices_card_le G)
theorem count_eq_zero_of_sparse_bound (G : Input)
(h : 2 * G.ordinary.records.length + G.loops.length < G.n) : count G = 0 := by
apply count_eq_zero_of_forall_weight_eq_zero
intro M
by_contra hM
exact (Nat.not_le_of_gt h) (vertex_bound_of_weight_ne_zero G M hM)
theorem vertex_bound_of_count_ne_zero (G : Input) (h : count G ≠ 0) :
G.n ≤ 2 * G.ordinary.records.length + G.loops.length := by
by_contra hb
exact h (count_eq_zero_of_sparse_bound G (Nat.lt_of_not_ge hb))
end LoopMatching
end OAI
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