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import OAI.Computability.VertexCover.Basic
import OAI.Computability.VertexCover.PCP.PreprocessingOverlayTables
import OAI.Computability.VertexCover.Analysis.GraphCompleteness
namespace OAI
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namespace VertexCover.ExplicitGraph
noncomputable section
def materialize {V : Type*} [Fintype V] (G : SimpleGraph V) : ExplicitGraph := by
classical
let e : V ≃ Fin (Fintype.card V) := Fintype.equivFin V
let E : Finset (Fin (Fintype.card V) × Fin (Fintype.card V)) :=
Finset.univ.filter (fun p => p.1 < p.2 ∧ G.Adj (e.symm p.1) (e.symm p.2))
exact ⟨Fintype.card V, E.toList,
fun p hp => (Finset.mem_filter.mp (Finset.mem_toList.mp hp)).2.1,
E.nodup_toList⟩
theorem materialize_n {V : Type*} [Fintype V] (G : SimpleGraph V) :
(materialize G).n = Fintype.card V := rfl
theorem materialize_mem_edges {V : Type*} [Fintype V] (G : SimpleGraph V)
(v w : Fin (Fintype.card V)) :
(v,w) ∈ (materialize G).edges ↔
v < w ∧ G.Adj ((Fintype.equivFin V).symm v) ((Fintype.equivFin V).symm w) := by
classical
change (v,w) ∈ (Finset.univ.filter (fun p : Fin (Fintype.card V) × Fin (Fintype.card V) =>
p.1 < p.2 ∧ G.Adj ((Fintype.equivFin V).symm p.1)
((Fintype.equivFin V).symm p.2))).toList ↔ _
exact Finset.mem_toList.trans (by simp only [Finset.mem_filter, Finset.mem_univ, true_and])
theorem coverNumber_le {G : ExplicitGraph} (C : Finset (Fin G.n)) (hC : G.Cover C) :
G.coverNumber ≤ C.card := Nat.find_min' _ ⟨C,hC,rfl⟩
theorem materialize_cover {V : Type*} [Fintype V] (G : SimpleGraph V)
(C : Finset V) (hC : G.IsVertexCover (C : Set V)) :
∃ S : Finset (Fin (materialize G).n), (materialize G).Cover S ∧ S.card = C.card := by
classical
let e := Fintype.equivFin V
refine ⟨C.map e.toEmbedding, ?_, Finset.card_map _⟩
intro p hp
have h := ((materialize_mem_edges G p.1 p.2).mp hp).2
have hh := hC h
rcases hh with hv | hw
· left
exact Finset.mem_map.mpr ⟨e.symm p.1, hv, e.apply_symm_apply _⟩
· right
exact Finset.mem_map.mpr ⟨e.symm p.2, hw, e.apply_symm_apply _⟩
theorem materialize_coverNumber_lt {V : Type*} [Fintype V] (G : SimpleGraph V)
(C : Finset V) (hC : G.IsVertexCover (C : Set V)) (ρ : ℝ)
(hcard : (C.card : ℝ) < ρ * Fintype.card V) :
((materialize G).coverNumber : ℝ) < ρ * (materialize G).n := by
obtain ⟨S, hS, hScard⟩ := materialize_cover G C hC
have hle := coverNumber_le S hS
rw [hScard] at hle
exact (Nat.cast_le.mpr hle).trans_lt hcard
end
end VertexCover.ExplicitGraph
namespace VertexCover.LabelCover
noncomputable section
def explicitGraph (Φ : LabelCover) (m : ℕ) : ExplicitGraph :=
ExplicitGraph.materialize (Φ.graph (Parameters.d m) (Parameters.t m))
theorem explicitGraph_vertices (Φ : LabelCover) (m : ℕ) :
(Φ.explicitGraph m).n = Φ.M^((Parameters.d m).choose 2) *
(2*Parameters.d m+1)^(Φ.WeightDimension (Parameters.d m) * Parameters.d m) :=
Φ.vertex_card _
theorem explicitGraph_complete (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m)
(A : Φ.Labeling) (hA : ∀ e, Φ.Satisfies A e) :
((Φ.explicitGraph m).coverNumber : ℝ) <
((1:ℝ)/2+1/(m:ℝ)) * (Φ.explicitGraph m).n := by
obtain ⟨C, hC, hcard⟩ := Φ.graph_completeness_cover m hm A hA
exact ExplicitGraph.materialize_coverNumber_lt _ C hC _ hcard
end
end VertexCover.LabelCover
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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