| import Mathlib |
| import OAI.Combinatorics.IndependentSets.Model |
| import OAI.Combinatorics.IndependentSets.Reduction.SamplerParameters |
|
|
| namespace OAI |
|
|
| namespace LargeIndependentSets |
| open scoped Classical BigOperators |
|
|
| namespace Graph |
|
|
| noncomputable def ofSimple {W : Type} [Fintype W] [Nonempty W] (H : SimpleGraph W) : Graph where |
| vertices := Fintype.card W |
| nonempty := Fintype.card_pos |
| adj u v := decide (H.Adj ((Fintype.equivFin W).symm u) ((Fintype.equivFin W).symm v)) |
| symm u v := by simp only [decide_eq_decide]; exact H.adj_comm _ _ |
| loopless u := by simp |
|
|
| theorem ofSimple_colorable {W : Type} [Fintype W] [Nonempty W] (H : SimpleGraph W) |
| (h : ∃ color : W → Fin 3, ∀ u v, H.Adj u v → color u ≠ color v) : |
| (ofSimple H).ThreeColorable := by |
| obtain ⟨color,hcolor⟩ := h |
| refine ⟨fun u => color ((Fintype.equivFin W).symm u),?_⟩ |
| intro u v h |
| exact hcolor _ _ (of_decide_eq_true h) |
|
|
| theorem ofSimple_independent {W : Type} [Fintype W] [Nonempty W] (H : SimpleGraph W) |
| (A : Finset (Fin (ofSimple H).vertices)) (hA : (ofSimple H).Independent A) : |
| H.IsIndepSet (A.image (Fintype.equivFin W).symm : Set W) := by |
| intro u hu v hv hne hadj |
| obtain ⟨a,ha,rfl⟩ := Finset.mem_image.mp hu |
| obtain ⟨b,hb,rfl⟩ := Finset.mem_image.mp hv |
| have hne' : a ≠ b := fun h => hne (congrArg _ h) |
| have hh := hA a ha b hb hne' |
| have ht : (ofSimple H).adj a b = true := by exact decide_eq_true hadj |
| exact Bool.false_ne_true (hh.symm.trans ht) |
|
|
| lemma exists_max_independent (G : Graph) : |
| ∃ A : Finset (Fin G.vertices), G.Independent A ∧ G.independenceNumber = A.card := by |
| let all := ((Finset.univ : Finset (Fin G.vertices)).powerset.filter G.Independent) |
| have hne : all.Nonempty := by |
| refine ⟨∅,?_⟩ |
| simp [all, Independent] |
| obtain ⟨A,hA,he⟩ := Finset.exists_mem_eq_sup all hne Finset.card |
| exact ⟨A,(Finset.mem_filter.mp hA).2,he⟩ |
|
|
| theorem ofSimple_alpha {W : Type} [Fintype W] [Nonempty W] (H : SimpleGraph W) |
| {δ : ℝ} (h : ∀ A : Finset W, H.IsIndepSet (A : Set W) → |
| (A.card : ℝ) / Fintype.card W < δ) : |
| ((ofSimple H).independenceNumber : ℝ) < δ * (ofSimple H).vertices := by |
| obtain ⟨A,hA,he⟩ := (ofSimple H).exists_max_independent |
| have hh := h (A.image (Fintype.equivFin W).symm) (ofSimple_independent H A hA) |
| have hc : (A.image (Fintype.equivFin W).symm).card = A.card := by |
| apply Finset.card_image_iff.mpr |
| exact (Fintype.equivFin W).symm.injective.injOn |
| rw [hc] at hh |
| rw [he] |
| exact (div_lt_iff₀ (Nat.cast_pos.mpr Fintype.card_pos)).mp hh |
|
|
| def clique (q : ℕ) (hq : 0 < q) : Graph where |
| vertices := q |
| nonempty := hq |
| adj u v := decide (u ≠ v) |
| symm u v := by simp [ne_comm] |
| loopless u := by simp |
|
|
| lemma clique_alpha_le (q : ℕ) (hq : 0 < q) : (clique q hq).independenceNumber ≤ 1 := by |
| obtain ⟨A,hA,he⟩ := (clique q hq).exists_max_independent |
| rw [he] |
| apply Finset.card_le_one.mpr |
| intro u hu v hv |
| by_contra hne |
| have hh := hA u hu v hv hne |
| let : DecidableEq (Fin q) := instDecidableEqFin q |
| have hh' : decide (u ≠ v) = false := hh |
| exact (of_decide_eq_false hh') hne |
|
|
| lemma clique_sound (q : ℕ) (hq : 0 < q) {δ : ℝ} (hδq : 1 < δ*q) : |
| ((clique q hq).independenceNumber : ℝ) < δ*(clique q hq).vertices := |
| lt_of_le_of_lt (by exact_mod_cast clique_alpha_le q hq) hδq |
|
|
| end Graph |
|
|
| namespace SamplerParameters |
|
|
| def CliqueTest (p : SamplerParameters) {U V L R C : Type} [Fintype L] [Fintype R] |
| (lc : LabelCoverData U V L R C) : Prop := |
| ∃ x : GridLocation (LayerAnswer L R (U:=U) (V:=V) (n:=p.n)) (p.m*2^p.k), |
| pathDistance (layeredSystem lc p.n).linkLength x (gridAntipode x) ≤ ENNReal.ofReal (1/8) |
|
|
| noncomputable def output (p : SamplerParameters) (q : ℕ) (hq : 0 < q) |
| {U V L R C : Type} [Fintype L] [Fintype R] [Fintype C] [Nonempty C] |
| (lc : LabelCoverData U V L R C) : Graph := |
| if p.CliqueTest lc then Graph.clique q hq else Graph.ofSimple (p.graph lc) |
|
|
| lemma output_complete (p : SamplerParameters) (q : ℕ) (hq : 0 < q) |
| {U V L R C : Type} [Fintype L] [Fintype R] [Fintype C] [Nonempty C] |
| (lc : LabelCoverData U V L R C) (ℓ : U → L) (ρ : V → R) (h : lc.Satisfied ℓ ρ) : |
| (p.output q hq lc).ThreeColorable := by |
| obtain ⟨hn,hcol⟩ := p.completeness lc ℓ ρ h |
| have ht : ¬p.CliqueTest lc := by |
| rintro ⟨x,hx⟩ |
| exact hn x hx |
| rw [output,ite_eq_right ht] |
| exact Graph.ofSimple_colorable _ hcol |
|
|
| lemma output_size (p : SamplerParameters) (q : ℕ) (hq : 0 < q) |
| {U V L R C : Type} [Fintype L] [Fintype R] [Fintype C] [Nonempty C] |
| (lc : LabelCoverData U V L R C) : |
| (p.output q hq lc).vertices ≤ max q (Fintype.card C ^ p.n * (p.law L R).denominator) := by |
| unfold output |
| split |
| · exact le_max_left _ _ |
| · change Fintype.card (p.Vertices L R C) ≤ _ |
| rw [RationalLaw.card_vertices] |
| exact le_max_right _ _ |
|
|
| end SamplerParameters |
|
|
| theorem finite_graph_reduction {δ : ℝ} (hδ : 0 < δ) : |
| ∃ p : SamplerParameters, ∃ σ : ℚ, ∃ q : ℕ, ∃ hq : 0 < q, |
| 0 < σ ∧ σ < 1 ∧ |
| ∀ (U V L R C : Type) [Fintype L] [Fintype R] [Fintype C] |
| [Nonempty L] [Nonempty R] [Nonempty C] (lc : LabelCoverData U V L R C), |
| ((∃ ℓ ρ, lc.Satisfied ℓ ρ) → (p.output q hq lc).ThreeColorable) ∧ |
| (lc.Sound (σ:ℝ) → ((p.output q hq lc).independenceNumber : ℝ) < |
| δ*(p.output q hq lc).vertices) := by |
| obtain ⟨p,σ,hσ,hσ1,H⟩ := finite_source_soundness hδ |
| obtain ⟨q,hq⟩ := exists_nat_gt (max (1:ℝ) (1/δ)) |
| have hq0 : 0 < q := by exact_mod_cast lt_trans (by norm_num : (0:ℝ) < 1) ((le_max_left _ _).trans_lt hq) |
| have hδq : 1 < δ*(q:ℝ) := by |
| have hh := (div_lt_iff₀ hδ).mp ((le_max_right _ _).trans_lt hq) |
| nlinarith |
| refine ⟨p,σ,q,hq0,hσ,hσ1,?_⟩ |
| intro U V L R C _ _ _ _ _ _ lc |
| constructor |
| · rintro ⟨ℓ,ρ,h⟩ |
| exact p.output_complete q hq0 lc ℓ ρ h |
| · intro hs |
| by_cases ht : p.CliqueTest lc |
| · rw [SamplerParameters.output,ite_eq_left ht] |
| exact Graph.clique_sound q hq0 hδq |
| · rw [SamplerParameters.output,ite_eq_right ht] |
| apply Graph.ofSimple_alpha |
| apply H U V L R C lc hs |
| intro x hx |
| exact ht ⟨x,hx⟩ |
|
|
| end LargeIndependentSets |
|
|
| end OAI |
|
|
