| import OAI.Combinatorics.CliqueFree.MultiplierLaw |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
| open Finset |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| namespace CliqueFreeIndependence.WeightedGraph |
| open Lottery |
| universe u |
| variable {V : Type u} [Fintype V] |
|
|
| / |
| zero outcome is inserted only at the end. -/ |
| lemma neighborhood_packing (G : SimpleGraph V) {w : V → ℝ} (hw : ∀ v, 0 < w v) |
| (A : Finset V) {s η b B : ℝ} (hs : 0 < s) (hη : 0 < η) (hb : 0 ≤ b) |
| (hB : 2 ≤ B) (hbig : 2/η*b ≤ B) (_hAs : mass w A ≤ s) |
| (hlocal : ∀ (R : Finset V), R ⊆ A → ∀ v ∈ R, |
| η*s < mass w (R ∩ neighbors G v) → |
| ∃ L : Law (V → ℝ), IsProb L ∧ BoundedOn L (R ∩ neighbors G v) b ∧ |
| (∀ u, expect L (fun m ↦ m u) = if u ∈ R ∩ neighbors G v then 1 else 0) ∧ |
| expectedEdges G w L ≤ η * (mass w (R ∩ neighbors G v))^2) : |
| ∃ L : Law (V → ℝ), Positive L ∧ total L ≤ 1/2+mass w A/(2*s) ∧ |
| BoundedOn L A B ∧ |
| (∀ u, expect L (fun m ↦ m u) = if u ∈ A then 1 else 0) ∧ |
| expectedEdges G w L ≤ 2*η*s*mass w A := by |
| classical |
| have hw0 := fun v ↦ (hw v).le |
| suffices hpack : ∀ R : Finset V, R ⊆ A → |
| ∃ L : Law (V → ℝ), Positive L ∧ total L ≤ 1/2+mass w R/(2*s) ∧ |
| BoundedOn L R B ∧ |
| (∀ u, expect L (fun m ↦ m u) = if u ∈ R then 1 else 0) ∧ |
| expectedEdges G w L ≤ 2*η*s*mass w R by exact hpack A Subset.rfl |
| intro R |
| refine Finset.strongInductionOn R ?_ |
| intro R ih hRA |
| by_cases hsmall : ∀ v ∈ R, mass w (R ∩ neighbors G v) ≤ η*s |
| · let m : V → ℝ := restrict R (fun _ ↦ 2) |
| let L : Law (V → ℝ) := [(1/2,m)] |
| refine ⟨L,?_,?_,?_,?_,?_⟩ |
| · intro z hz |
| have hz' : z = (1/2,m) := by simpa [L] using hz |
| rw [hz']; norm_num |
| · have hr := div_nonneg (mass_nonneg hw0 R) (by positivity : 0 ≤ 2*s) |
| simp only [L,total,expect_singleton,mul_one] |
| linarith |
| · intro z hz |
| have hz' : z = (1/2,m) := by simpa [L] using hz |
| rw [hz'] |
| refine ⟨?_,?_⟩ |
| · intro v |
| dsimp [m,restrict] |
| split_ifs <;> constructor <;> linarith |
| · intro v hv |
| simp [m,restrict,hv] |
| · intro v |
| simp [L,m,restrict] |
| · have he := edgeMass_restrict_le G hw0 R hsmall |
| have heq : (fun v ↦ w v*m v) = fun v ↦ 2*restrict R w v := by |
| funext v |
| by_cases hv : v ∈ R <;> simp [m,restrict,hv,mul_comm] |
| simp only [L,expectedEdges,expect_singleton,heq,edgeMass_smul] |
| have hn := mul_nonneg (mul_nonneg hη.le hs.le) (mass_nonneg hw0 R) |
| nlinarith |
| · push Not at hsmall |
| obtain ⟨v,hv,hvlarge⟩ := hsmall |
| let D := R ∩ neighbors G v |
| have hDR : D ⊆ R := inter_subset_left |
| have hDpos : 0 < mass w D := (mul_pos hη hs).trans hvlarge |
| have hDne : D.Nonempty := by |
| by_contra hn |
| have he : D = ∅ := not_nonempty_iff_eq_empty.1 hn |
| simp [he,mass] at hDpos |
| obtain ⟨L,hLp,hLb,hLm,hLe⟩ := hlocal R hRA v hv hvlarge |
| obtain ⟨K,hKp,hKt,hKb,hKm,hKe⟩ := ih (R \ D) (sdiff_ssubset hDR hDne) (sdiff_subset.trans hRA) |
| let p := mass w D / (2*s) |
| have hp : 0 < p := div_pos hDpos (by positivity) |
| have hpη : η/2 < p := by |
| apply (lt_div_iff₀ (by positivity : 0 < 2*s)).2 |
| nlinarith only [hvlarge] |
| have hik : (1/p)*b ≤ B := by |
| have hi : 1/p ≤ 2/η := by |
| apply (div_le_div_iff₀ hp hη).2 |
| linarith |
| exact (mul_le_mul_of_nonneg_right hi hb).trans hbig |
| let J := transform p (fun m u ↦ (1/p)*m u) L |
| have hJp : Positive J := positive_transform hp.le _ hLp.1 |
| have hJb : BoundedOn J D B := bounded_transform hLb (one_div_nonneg.2 hp.le) hik p |
| have hpart : mass w (R \ D) + mass w D = mass w R := by |
| exact sum_sdiff hDR |
| have hJt : total J = p := by |
| rw [total_transform,hLp.2,mul_one] |
| have hJm (u : V) : expect J (fun m ↦ m u) = if u ∈ D then 1 else 0 := by |
| rw [mean_transform,hLm,mul_one_div_cancel hp.ne',one_mul] |
| have hJe : expectedEdges G w J ≤ 2*η*s*mass w D := by |
| rw [expectedEdges_transform] |
| have he := mul_le_mul_of_nonneg_left hLe (by positivity : 0 ≤ p*(1/p)^2) |
| have hid : p*(1/p)^2*(η*(mass w D)^2) = 2*η*s*mass w D := by |
| dsimp [p] |
| field_simp |
| exact he.trans_eq hid |
| refine ⟨J++K,positive_append hJp hKp,?_,bounded_append hJb hKb hDR sdiff_subset,?_,?_⟩ |
| · change expect (J++K) (fun _ ↦ 1) ≤ _ |
| rw [expect_append] |
| change total J + total K ≤ _ |
| rw [hJt] |
| dsimp [p] |
| have hparts : mass w D/(2*s) + mass w (R \ D)/(2*s) = mass w R/(2*s) := by |
| rw [← add_div] |
| congr 1 |
| linarith only [hpart] |
| linarith only [hKt,hparts] |
| · intro u |
| rw [expect_append,hJm,hKm] |
| by_cases hu : u ∈ D |
| · have huR := hDR hu |
| simp [hu,huR] |
| · by_cases huR : u ∈ R <;> simp [hu,huR] |
| · change expect (J++K) _ ≤ _ |
| rw [expect_append] |
| change expectedEdges G w J + expectedEdges G w K ≤ _ |
| calc |
| _ ≤ 2*η*s*mass w D + 2*η*s*mass w (R \ D) := add_le_add hJe hKe |
| _ = _ := by rw [← mul_add,add_comm (mass w D),hpart] |
|
|
| end CliqueFreeIndependence.WeightedGraph |
|
|
| end |
|
|
| end OAI |
|
|
