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97c3405 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 | 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]
/-- Greedy disjoint-neighborhood packing, with a subprobability lottery. The additional
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
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