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import OAI.Computability.VertexCover.Reduction.Soundness
import OAI.Computability.VertexCover.Reduction.GraphSoundness
namespace OAI
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namespace VertexCover.ExplicitGraph
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
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
namespace VertexCover
noncomputable section
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))
end
end VertexCover
end
section
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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