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import OAI.Computability.VertexCover.Analysis.DenseSeparatorData
import OAI.Computability.VertexCover.Reduction.RestrictionFrozen
namespace OAI
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namespace VertexCover.Parameters
theorem h_ge_three (m : ℕ) (hm : 0 < m) : 3 ≤ h m := by
have hpow : 1 ≤ m^3 := one_le_pow₀ hm
unfold h
norm_num only [Nat.reducePow]
omega
theorem private_variance_threshold (m : ℕ) (hm : 0 < m) : ν m*(h m:ℝ)/4 = 8 := by
have hn : (m:ℝ) ≠ 0 := by positivity
simp only [ν, h, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat]
field_simp
norm_num
end VertexCover.Parameters
namespace VertexCover.LabelCover
open VertexCover.Restriction
theorem dense_privateVariance_lower (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m)
(A : Finset (Φ.Vertex (Parameters.d m)))
(hA : (Φ.graph (Parameters.d m) (Parameters.t m)).IsIndepSet
(A : Set (Φ.Vertex (Parameters.d m))))
(hdense : (Fintype.card (Φ.Vertex (Parameters.d m)):ℝ)/(m:ℝ) ≤ A.card) :
∃ hV : (Φ.highVectors (Parameters.t m) A).Nonempty,
∃ J : Finset (Fin (Parameters.d m)), J.card = Parameters.h m ∧
∃ frozen : Φ.Seeds (Parameters.d m),
∃ weights : Fin (Parameters.d m) → Fin (Φ.WeightDimension (Parameters.d m)) → ℝ,
Φ.InWeightCube weights ∧
8 ≤ Φ.ownTotalVariance J frozen (Φ.outsideSum J frozen weights)
(Φ.highVectors (Parameters.t m) A) hV := by
obtain ⟨hV, _hzero, henergy⟩ := Φ.dense_separator_data m hm A hA hdense
rw [← Φ.restrictionFunction_energy_variance] at henergy
have hdim := Parameters.dimensions m hm
obtain ⟨J, hJ, frozen, s, hs⟩ := Φ.restriction_frozen_exists
hdim.1 (Parameters.h_ge_three m (by omega)) hdim.2.2.1.le
(Parameters.ν m) (Parameters.positive m (by omega)).1
(Parameters.restriction_budget m hm) _ hV henergy
refine ⟨hV, J, hJ, frozen, CompactCube.realWeights s, ?_, ?_⟩
· intro j k
exact abs_le.mpr (s j k).2
· simpa only [Parameters.private_variance_threshold m (by omega)] using hs
theorem graph_soundness_independent (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m)
(hval : Φ.value ≤ Parameters.σ m)
(A : Finset (Φ.Vertex (Parameters.d m)))
(hA : (Φ.graph (Parameters.d m) (Parameters.t m)).IsIndepSet
(A : Set (Φ.Vertex (Parameters.d m)))) :
(A.card:ℝ) < (Fintype.card (Φ.Vertex (Parameters.d m)):ℝ)/(m:ℝ) := by
by_contra hn
obtain ⟨hV, J, hJ, frozen, weights, _hw, hlow⟩ :=
Φ.dense_privateVariance_lower m hm A hA (le_of_not_gt hn)
have hu := Φ.ownTotalVariance_no_upper m hm hval J hJ frozen
(Φ.outsideSum J frozen weights) _ hV
linarith
theorem graph_soundness_cover (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m)
(hval : Φ.value ≤ Parameters.σ m)
(C : Finset (Φ.Vertex (Parameters.d m)))
(hC : (Φ.graph (Parameters.d m) (Parameters.t m)).IsVertexCover
(C : Set (Φ.Vertex (Parameters.d m)))) :
(1-1/(m:ℝ)) * Fintype.card (Φ.Vertex (Parameters.d m)) < (C.card:ℝ) := by
classical
have hI : (Φ.graph (Parameters.d m) (Parameters.t m)).IsIndepSet
(↑(Cᶜ) : Set (Φ.Vertex (Parameters.d m))) := by
simpa only [Finset.coe_compl] using SimpleGraph.isIndepSet_compl_iff_isVertexCover.mpr hC
have hi := Φ.graph_soundness_independent m hm hval Cᶜ hI
rw [Finset.card_compl, Nat.cast_sub (Finset.card_le_univ C)] at hi
simp only [div_eq_mul_inv, one_mul] at hi ⊢
nlinarith
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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