| import Mathlib |
| import OAI.Computability.VertexCover.Reduction.Soundness |
| import OAI.Computability.VertexCover.Reduction.GraphSoundness |
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| namespace OAI |
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| namespace VertexCover.ExplicitGraph |
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|
| theorem coverNumber_witness (G : ExplicitGraph) : |
| ∃ S : Finset (Fin G.n), G.Cover S ∧ S.card = G.coverNumber := by |
| have hex : ∃ k : ℕ, ∃ S : Finset (Fin G.n), G.Cover S ∧ S.card = k := |
| ⟨G.n, Finset.univ, by simp [Cover], by simp⟩ |
| exact Nat.find_spec hex |
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|
| theorem materialize_cover_reflects {V : Type*} [Fintype V] (G : SimpleGraph V) |
| (S : Finset (Fin (materialize G).n)) (hS : (materialize G).Cover S) : |
| ∃ C : Finset V, G.IsVertexCover (C : Set V) ∧ C.card = S.card := by |
| classical |
| let e := Fintype.equivFin V |
| let C := S.map e.symm.toEmbedding |
| have hmem (v : V) : v ∈ C ↔ e v ∈ S := by |
| exact Finset.mem_map_equiv |
| refine ⟨C, ?_, Finset.card_map _⟩ |
| intro v w hvw |
| have hne : e v ≠ e w := fun he => G.ne_of_adj hvw (e.injective he) |
| rcases lt_or_gt_of_ne hne with hlt | hgt |
| · have hp := hS (e v, e w) ((materialize_mem_edges G (e v) (e w)).mpr |
| ⟨hlt, by simpa only [e, Equiv.symm_apply_apply] using hvw⟩) |
| exact hp.imp (hmem v).mpr (hmem w).mpr |
| · have hp := hS (e w, e v) ((materialize_mem_edges G (e w) (e v)).mpr |
| ⟨hgt, by simpa only [e, Equiv.symm_apply_apply] using hvw.symm⟩) |
| exact hp.symm.imp (hmem v).mpr (hmem w).mpr |
|
|
| theorem materialize_coverNumber_gt {V : Type*} [Fintype V] (G : SimpleGraph V) |
| (ρ : ℝ) (h : ∀ C : Finset V, G.IsVertexCover (C : Set V) → |
| ρ * Fintype.card V < (C.card:ℝ)) : |
| ρ * (materialize G).n < ((materialize G).coverNumber:ℝ) := by |
| obtain ⟨S, hS, hmin⟩ := coverNumber_witness (materialize G) |
| obtain ⟨C, hC, hcard⟩ := materialize_cover_reflects G S hS |
| have hc := h C hC |
| rw [hcard, hmin] at hc |
| exact hc |
|
|
| end VertexCover.ExplicitGraph |
|
|
| namespace VertexCover.LabelCover |
|
|
| theorem explicitGraph_sound (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m) |
| (hval : Φ.value ≤ Parameters.σ m) : |
| (1-1/(m:ℝ))*(Φ.explicitGraph m).n < ((Φ.explicitGraph m).coverNumber:ℝ) := by |
| exact ExplicitGraph.materialize_coverNumber_gt _ _ (Φ.graph_soundness_cover m hm hval) |
|
|
| end VertexCover.LabelCover |
|
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| namespace VertexCover |
| noncomputable section |
|
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| theorem rawGraph_soundness (m : ℕ) (hm : 4 ≤ m) (input : List Bool) |
| (h : ¬ ThreeSAT input) : |
| (1 - 1 / (m : ℝ)) * (rawGraph m hm input).n < |
| ((rawGraph m hm input).coverNumber : ℝ) := by |
| exact LabelCover.explicitGraph_sound _ m hm |
| (le_of_lt (rawLabelCover_soundness m hm input h)) |
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| end |
| end VertexCover |
| end |
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| end OAI |
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