| import OAI.Probability.MatroidProphet.Main |
|
|
| namespace OAI |
|
|
| /-! |
| Source-facing contracts for the candidate filter (setup.tex, Lemma `lem:filter`). |
| The sets below are the actual rounded-priority greedy survivors and candidate |
| labels, not arbitrary sets satisfying a desired inequality. The fair-mask law |
| is the concrete product law `bitsExpectation` on the entire labeled ground set. |
| -/ |
|
|
| namespace MatroidProphet.CandidateContracts |
|
|
| open Finset MainAlgorithm |
|
|
| /-- The full candidate-mass contract, including pointwise independence and |
| containment for every mask. No positivity premise is needed here because the |
| rounding map itself discards nonpositive coordinates. -/ |
| theorem candidate_mass {n : ℕ} (M : Matroid (Fin n)) |
| (hE : M.E = Set.univ) (w : Weights n) : |
| let a := fun e => roundedLevel weightBase (w e) |
| let Y := fun H => positiveGreedy M a \ H |
| let U := candidateLabels M a |
| (∀ H, M.Indep (Y H : Set (Fin n)) ∧ Y H ⊆ U H) ∧ |
| bitsExpectation (fun _ => (1 / 2 : ℝ)) Finset.univ |
| (fun H => ∑ e ∈ Y H, roundedWeight weightBase (w e)) = |
| optimum M (fun e => roundedWeight weightBase (w e)) / 2 ∧ |
| bitsExpectation (fun _ => (1 / 2 : ℝ)) Finset.univ |
| (fun H => ∑ e ∈ U H, roundedWeight weightBase (w e)) ≤ |
| optimum M (fun e => roundedWeight weightBase (w e)) := by |
| dsimp only |
| refine ⟨?_, ?_, ?_⟩ |
| · intro H |
| exact ⟨positiveGreedy_survivors_indep M hE _ H, |
| positiveGreedy_survivors_subset M _ H⟩ |
| · exact expectation_positiveGreedy_survivors M hE w (by norm_num [weightBase]) |
| · exact expectation_candidateLabels_le_optimum M hE w weightBase |
|
|
| /-- Both inequalities of the source's rounding bound, together with an actual |
| independent positive-priority greedy maximizer. This includes zero weights, |
| loops, ties, the empty ground set, and rank zero. -/ |
| theorem rounding_and_greedy {n : ℕ} (M : Matroid (Fin n)) |
| (hE : M.E = Set.univ) (w : Weights n) (hw : ∀ e, 0 ≤ w e) : |
| let u := fun e => roundedWeight weightBase (w e) |
| let I := positiveGreedy M (fun e => roundedLevel weightBase (w e)) |
| optimum M w / weightBase ≤ optimum M u ∧ |
| optimum M u ≤ optimum M w ∧ |
| M.Indep (I : Set (Fin n)) ∧ |
| (∑ e ∈ I, u e) = optimum M u := by |
| dsimp only |
| have hB : (1 : ℝ) < weightBase := by norm_num [weightBase] |
| refine ⟨?_, ?_, positiveGreedy_indep M hE _, ?_⟩ |
| · apply (div_le_iff₀ (zero_lt_one.trans hB)).2 |
| simpa only [mul_comm] using optimum_rounding_bound M weightBase hB w hw |
| · exact optimum_mono M _ w (fun e => roundedWeight_le hB (hw e)) |
| · change (∑ e ∈ positiveGreedy M (fun e => roundedLevel weightBase (w e)), |
| levelWeight weightBase (roundedLevel weightBase (w e))) = _ |
| rw [sum_positiveGreedy] |
| exact Priority.rounded_greedy_optimal M hE w hB |
|
|
| /-- The actual candidate predicate never includes a loop or a nonpositive label. |
| This is a pointwise statement for every mask, not only almost surely. -/ |
| theorem candidate_positive_nonloop {n : ℕ} (M : Matroid (Fin n)) |
| (w : Weights n) (H : Finset (Fin n)) (e : Fin n) |
| (he : e ∈ candidateLabels M (fun f => roundedLevel weightBase (w f)) H) : |
| 0 < w e ∧ e ∉ H ∧ e ∉ M.closure ∅ := by |
| classical |
| have hc : candidate M ⟨H, ∅, ∅, ∅, false⟩ |
| (fun f => roundedLevel weightBase (w f)) e (roundedLevel weightBase (w e)) := |
| (Finset.mem_filter.mp he).2 |
| refine ⟨?_, hc.2.1, candidate_not_mem_closure_empty M _ _ _ _ hc⟩ |
| by_contra hn |
| exact hc.1 (by simp [roundedLevel, hn]) |
|
|
| end MatroidProphet.CandidateContracts |
|
|
| end OAI |
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|
