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dd14c26 | 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 | import OAI.Probability.MatroidProphet.WeightGroups
import OAI.Probability.MatroidProphet.WeightedPayoff
import OAI.Probability.MatroidProphet.Maximum.Score
namespace OAI
namespace MatroidProphet.MainAlgorithm
open Finset
variable {n : ℕ}
noncomputable def trueLevels (M : Matroid (Fin n)) (a : Fin n → Option ℤ)
(H : Finset (Fin n)) : Finset ℤ := weightLevels a (candidateLabels M a H)
noncomputable def analyzedLevels (M : Matroid (Fin n)) (a : Fin n → Option ℤ)
(H : Finset (Fin n)) : Finset ℤ := by
classical
exact (trueLevels M a H).filter fun i =>
densityThreshold ≤ ((candidateLabels M a H).filter (fun e => a e = some i)).card
noncomputable def comparisonMass (M : Matroid (Fin n)) (a : Fin n → Option ℤ)
(H : Finset (Fin n)) : ℝ :=
∑ e ∈ positiveGreedy M a \ H, levelWeight weightBase (a e)
noncomputable def analyzedComparisonMass (M : Matroid (Fin n)) (a : Fin n → Option ℤ)
(H : Finset (Fin n)) : ℝ := by
classical
exact ∑ i ∈ analyzedLevels M a H, weightBase ^ i *
(((positiveGreedy M a \ H).filter (fun e => a e = some i)).card : ℝ)
lemma candidate_level_le_maximumRounded (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(w : Weights n) (H : Finset (Fin n)) (i : ℤ)
(hi : i ∈ trueLevels M (fun e => roundedLevel weightBase (w e)) H) :
weightBase ^ i ≤ maximumRounded M w := by
classical
obtain ⟨e, he, hei⟩ := (mem_weightLevels _ _ i).1 hi
have hc : candidate M ⟨H, ∅, ∅, ∅, false⟩ (fun e => roundedLevel weightBase (w e)) e
(roundedLevel weightBase (w e)) := (Finset.mem_filter.mp he).2
have heI := candidate_singleton_indep M hE _ _ e _ hc
have h := roundedWeight_le_maximumRounded M w e heI
simpa only [roundedWeight, hei, levelWeight] using h
lemma comparisonMass_partition (M : Matroid (Fin n)) (a : Fin n → Option ℤ)
(H : Finset (Fin n)) :
comparisonMass M a H =
∑ i ∈ trueLevels M a H, weightBase ^ i *
(((positiveGreedy M a \ H).filter (fun e => a e = some i)).card : ℝ) := by
exact (sum_weightGroups_mass a _ _
(weightLevels_mono a (positiveGreedy_survivors_subset M a H)) weightBase).symm
lemma small_comparison_mass (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(w : Weights n) (H : Finset (Fin n)) :
comparisonMass M (fun e => roundedLevel weightBase (w e)) H ≤
analyzedComparisonMass M (fun e => roundedLevel weightBase (w e)) H +
2 * densityThreshold * maximumRounded M w := by
classical
let a : Fin n → Option ℤ := fun e => roundedLevel weightBase (w e)
let Y := positiveGreedy M a \ H
let U := candidateLabels M a H
let L := trueLevels M a H
let A := analyzedLevels M a H
have hAU : A ⊆ L := Finset.filter_subset _ _
have hsmall : ∀ i ∈ L \ A, ((Y.filter (fun e => a e = some i)).card : ℝ) ≤ densityThreshold := by
intro i hi
obtain ⟨hiL, hiA⟩ := Finset.mem_sdiff.mp hi
have hn : ¬densityThreshold ≤ ((U.filter (fun e => a e = some i)).card : ℝ) := by
intro hh
exact hiA (Finset.mem_filter.mpr ⟨hiL, hh⟩)
have hc : (Y.filter (fun e => a e = some i)).card ≤ (U.filter (fun e => a e = some i)).card :=
Finset.card_le_card (Finset.filter_subset_filter _ (positiveGreedy_survivors_subset M a H))
have hcR : ((Y.filter (fun e => a e = some i)).card : ℝ) ≤
((U.filter (fun e => a e = some i)).card : ℝ) := by exact_mod_cast hc
exact hcR.trans (le_of_lt (lt_of_not_ge hn))
have hbound := small_group_mass_bound weightBase densityThreshold (maximumRounded M w)
(by norm_num [weightBase]) constants_positive.2.1.le (maximumRounded_nonneg M w)
(L \ A) (fun i => ((Y.filter (fun e => a e = some i)).card : ℝ)) hsmall
(fun i hi => candidate_level_le_maximumRounded M hE w H i (Finset.mem_sdiff.mp hi).1)
have hpartition := Finset.sum_sdiff hAU
(f := fun i => weightBase ^ i * ((Y.filter (fun e => a e = some i)).card : ℝ))
rw [comparisonMass_partition]
change (∑ i ∈ L, weightBase ^ i * ((Y.filter (fun e => a e = some i)).card : ℝ)) ≤
(∑ i ∈ A, weightBase ^ i * ((Y.filter (fun e => a e = some i)).card : ℝ)) + _
linarith
end MatroidProphet.MainAlgorithm
end OAI
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