| import OAI.Combinatorics.CliqueFree.VertexSplitting |
| import OAI.Combinatorics.CliqueFree.Peeling |
|
|
| namespace OAI |
|
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| noncomputable section |
|
|
| open scoped BigOperators |
| open Finset |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| namespace CliqueFreeIndependence.WeightedGraph |
| universe u |
| variable {V : Type u} [Fintype V] |
|
|
| omit [Fintype V] in |
| lemma EdgeProjection.of_le {G H : SimpleGraph V} (hHG : H ≤ G) : EdgeProjection H G id where |
| map_adj := fun _ _ h ↦ hHG h |
| inj := fun _ _ _ _ _ _ h1 h2 ↦ ⟨h1,h2⟩ |
|
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| lemma neighborMass_bot (w : V → ℝ) (u : V) : neighborMass (⊥ : SimpleGraph V) w u = 0 := by |
| simp [neighborMass, neighbors, mass] |
|
|
| lemma triangleMass_bot (w : V → ℝ) : triangleMass (⊥ : SimpleGraph V) w = 0 := by |
| simp [triangleMass,triangleAt,neighbors] |
|
|
| lemma splittingConstant_nonneg {C : ℝ} (hC : 1 ≤ C) : 0 ≤ splittingConstant C := by |
| unfold splittingConstant smoothingConstant entropyConstant exceptionalConstant |
| positivity |
|
|
| / |
| lemma neighborhood_reduction (G : SimpleGraph V) {w : V → ℝ} (hw : ∀ u, 0 < w u) |
| {C x : ℝ} (hC : 1 ≤ C) (hCross : CrossBound G w C) (hx : 32 ≤ x) |
| (hε : splittingConstant C * Numerical.lossRatio x ≤ 1) |
| (hL : ∀ u, neighborMass G w u ≤ Real.exp x) : |
| ∃ (U : Type u) (_ : Fintype U) (J : SimpleGraph U) (π : U → V), |
| EdgeProjection J G π ∧ (∀ a, neighborMass J (w ∘ π) a ≤ Real.exp (Real.sqrt x)) ∧ |
| (1-splittingConstant C * Numerical.lossRatio x) * triangleMass G w - edgeMass G w / x ≤ |
| triangleMass J (w ∘ π) := by |
| have hx0 : 0 < x := by linarith |
| have hw0 := fun v ↦ (hw v).le |
| have hε0 : 0 ≤ splittingConstant C * Numerical.lossRatio x := |
| mul_nonneg (splittingConstant_nonneg hC) (Numerical.lossRatio_nonneg (by linarith)) |
| obtain ⟨H,hHG,hc,hpeel⟩ := peeling G hw (one_div_pos.2 hx0) |
| have hprojH := EdgeProjection.of_le hHG |
| have hcH : CrossBound H w C := by |
| simpa only [Function.comp_id] using hprojH.crossBound hw0 (by linarith) hCross |
| have hLH : ∀ u, neighborMass H w u ≤ Real.exp x := by |
| intro u |
| exact (hprojH.neighborMass_le hw0 u).trans (hL u) |
| by_cases hne : Nonempty (EdgeState H) |
| · let := hne |
| obtain ⟨J,hproj,hJ,hT⟩ := weighted_splitting hw hC hcH hx hc hLH |
| refine ⟨EdgeState H,inferInstance,J,EdgeState.src,?_,hJ,?_⟩ |
| · exact ⟨fun a b h ↦ hHG (hproj.map_adj a b h),hproj.inj⟩ |
| · change (1-splittingConstant C * Numerical.lossRatio x) * triangleMass H w ≤ |
| triangleMass J (w ∘ EdgeState.src) at hT |
| have hh := mul_le_mul_of_nonneg_left hpeel (sub_nonneg.2 hε) |
| have hm := div_nonneg (edgeMass_nonneg G hw0) hx0.le |
| rw [one_div_mul_eq_div] at hh |
| nlinarith only [hT, hh, mul_nonneg hε0 hm] |
| · have hbot : H = ⊥ := by |
| apply SimpleGraph.ext |
| funext a b |
| apply propext |
| simpa only [SimpleGraph.bot_adj, iff_false] using |
| (fun hab : H.Adj a b ↦ hne ⟨⟨(a,b),hab⟩⟩) |
| rw [hbot,triangleMass_bot] at hpeel |
| refine ⟨V,inferInstance,⊥,id,EdgeProjection.of_le bot_le,?_,?_⟩ |
| · intro a |
| simp only [Function.comp_id,neighborMass_bot] |
| exact (Real.exp_pos _).le |
| · simp only [Function.comp_id,triangleMass_bot] |
| rw [one_div_mul_eq_div] at hpeel |
| nlinarith [mul_nonneg hε0 (triangleMass_nonneg G hw0)] |
|
|
| / |
| lemma triangle_bound_scale {C X : ℝ} (hC : 1 ≤ C) (hX : 32 ≤ X) |
| (hsmall : splittingConstant C * Numerical.lossRatio X ≤ 1/4) (n : ℕ) : |
| ∀ {V : Type u} [Fintype V] (G : SimpleGraph V) (w : V → ℝ), |
| (∀ u, 0 < w u) → CrossBound G w C → |
| (∀ u, neighborMass G w u ≤ Real.exp (Numerical.scale X n)) → |
| triangleMass G w ≤ (1+Real.exp X/3) * (2-(1/2:ℝ)^n) * edgeMass G w := by |
| induction n with |
| | zero => |
| intro V _ G w hw hCross hL |
| have ht := triangleMass_le_of_neighborMass G (fun v ↦ (hw v).le) hL |
| simp only [Numerical.scale, pow_zero] at ht ⊢ |
| have he := edgeMass_nonneg G (fun v ↦ (hw v).le) |
| nlinarith |
| | succ n ih => |
| intro V _ G w hw hCross hL |
| let x := Numerical.scale X (n+1) |
| let q : ℝ := (1/2:ℝ)^(n+1) |
| let ε := splittingConstant C * Numerical.lossRatio x |
| have hx : 32 ≤ x := hX.trans (Numerical.scale_ge (by linarith) _) |
| have hq0 : 0 ≤ q := by positivity |
| have hq1 : q ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num) |
| have hε0 : 0 ≤ ε := mul_nonneg (splittingConstant_nonneg hC) |
| (Numerical.lossRatio_nonneg (by linarith)) |
| have hε : ε ≤ q/4 := by |
| have hh := mul_le_mul_of_nonneg_left (Numerical.lossRatio_scale (by linarith : 16 ≤ X) (n+1)) |
| (splittingConstant_nonneg hC) |
| have hh2 := mul_le_mul_of_nonneg_left hsmall hq0 |
| dsimp [ε,x,q] |
| nlinarith only [hh,hh2] |
| obtain ⟨U,inst,J,π,hproj,hJ,hstep⟩ := neighborhood_reduction G hw hC hCross hx |
| (by linarith : ε ≤ 1) hL |
| let := inst |
| have hsqrt : Real.sqrt x = Numerical.scale X n := by |
| dsimp [x,Numerical.scale] |
| exact Real.sqrt_sq (by linarith [Numerical.scale_ge (by linarith : 1 ≤ X) n]) |
| rw [hsqrt] at hJ |
| have hJcross := hproj.crossBound (fun v ↦ (hw v).le) (by linarith) hCross |
| have hupper := ih J (w ∘ π) (fun a ↦ hw (π a)) hJcross hJ |
| have hmass := hproj.edgeMass_le (fun v ↦ (hw v).le) |
| have hcoeff : 0 ≤ (1+Real.exp X/3)*(2-(1/2:ℝ)^n) := by |
| apply mul_nonneg (by positivity) |
| have hp : (1/2:ℝ)^n ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num) |
| linarith |
| have hupperG := hupper.trans (mul_le_mul_of_nonneg_left hmass hcoeff) |
| have hqeq : (1/2:ℝ)^n = 2*q := by dsimp [q]; rw [pow_succ]; ring |
| rw [hqeq] at hupperG |
| have hinv : 1/x ≤ q/2 := by |
| have hi := Numerical.inv_scale (by linarith : 2 ≤ X) (n+1) |
| have hXi : 1/X ≤ 1/2 := (one_div_le_one_div_of_le (by norm_num) (by linarith)) |
| have hi2 := mul_le_mul_of_nonneg_left hXi hq0 |
| dsimp [x,q] at * |
| linarith |
| apply Numerical.recurrence_bound (by linarith [Real.exp_pos X] : 1 ≤ 1+Real.exp X/3) |
| hq0 hq1 hε0 hε (triangleMass_nonneg G (fun v ↦ (hw v).le)) |
| (edgeMass_nonneg G (fun v ↦ (hw v).le)) hinv |
| calc |
| (1-ε)*triangleMass G w - (1/x)*edgeMass G w = |
| (1-ε)*triangleMass G w - edgeMass G w/x := by ring |
| _ ≤ triangleMass J (w ∘ π) := hstep |
| _ ≤ _ := hupperG |
|
|
| / |
| theorem weighted_triangles {C : ℝ} (hC : 1 ≤ C) : |
| ∃ B : ℝ, 0 < B ∧ ∀ {V : Type u} [Fintype V] (G : SimpleGraph V) (w : V → ℝ), |
| (∀ v, 0 < w v) → CrossBound G w C → triangleMass G w ≤ B * edgeMass G w := by |
| obtain ⟨X,hX,hsmall⟩ := Numerical.exists_iteration_threshold (splittingConstant C) |
| refine ⟨2*(1+Real.exp X/3),by positivity,?_⟩ |
| intro V _ G w hw hCross |
| obtain ⟨n,hn⟩ := exists_nat_ge (mass w (univ : Finset V)) |
| have hL : ∀ v, neighborMass G w v ≤ Real.exp (Numerical.scale X n) := by |
| intro v |
| calc |
| _ ≤ mass w univ := mass_mono (fun v ↦ (hw v).le) (subset_univ _) |
| _ ≤ n := hn |
| _ ≤ Numerical.scale X n := by |
| have hh := Numerical.scale_ge_add (by linarith : 2 ≤ X) n |
| linarith |
| _ ≤ Real.exp (Numerical.scale X n) := by linarith [Real.add_one_le_exp (Numerical.scale X n)] |
| have ht := triangle_bound_scale hC hX hsmall n G w hw hCross hL |
| have he := edgeMass_nonneg G (fun v ↦ (hw v).le) |
| have hB : 0 ≤ 1+Real.exp X/3 := by positivity |
| have hq : 0 ≤ (1/2:ℝ)^n := by positivity |
| nlinarith [mul_nonneg hB hq, mul_nonneg (mul_nonneg hB hq) he] |
|
|
| end CliqueFreeIndependence.WeightedGraph |
|
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| end |
|
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| end OAI |
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