| import OAI.Probability.MatroidProphet.Main |
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| namespace OAI |
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| namespace MatroidProphet.SourceContracts |
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| open Finset Set MainAlgorithm |
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| / |
| theorem roundedWeight_pos_iff {B x : ℝ} (hB : 0 < B) : |
| 0 < roundedWeight B x ↔ 0 < x := by |
| unfold roundedWeight roundedLevel |
| by_cases hx : 0 < x |
| · rw [ite_eq_left hx, levelWeight] |
| exact iff_of_true (zpow_pos hB _) hx |
| · rw [ite_eq_right hx, levelWeight] |
| exact iff_of_false (lt_irrefl 0) hx |
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| / |
| levels taking priority and the original fixed label order breaking ties. -/ |
| theorem mem_candidateLabels_iff {n : ℕ} (M : Matroid (Fin n)) |
| (w : Weights n) (B : ℝ) (H : Finset (Fin n)) (e : Fin n) : |
| e ∈ candidateLabels M (fun f => roundedLevel B (w f)) H ↔ |
| 0 < w e ∧ e ∉ H ∧ |
| e ∉ M.closure {f | f ∈ H ∧ |
| higher f (roundedLevel B (w f)) e (roundedLevel B (w e))} := by |
| classical |
| have hpos : roundedLevel B (w e) ≠ none ↔ 0 < w e := by |
| simp [roundedLevel] |
| constructor |
| · intro hcandidate |
| obtain ⟨hlevel, houtside, hclosure⟩ := (Finset.mem_filter.mp hcandidate).2 |
| exact ⟨hpos.mp hlevel, houtside, hclosure⟩ |
| · rintro ⟨hpositive, houtside, hclosure⟩ |
| exact Finset.mem_filter.mpr |
| ⟨Finset.mem_univ _, hpos.mpr hpositive, houtside, hclosure⟩ |
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| / |
| theorem mem_candidateLabels_iff_rounded {n : ℕ} (M : Matroid (Fin n)) |
| (w : Weights n) {B : ℝ} (hB : 0 < B) (H : Finset (Fin n)) (e : Fin n) : |
| e ∈ candidateLabels M (fun f => roundedLevel B (w f)) H ↔ |
| 0 < roundedWeight B (w e) ∧ e ∉ H ∧ |
| e ∉ M.closure {f | f ∈ H ∧ |
| higher f (roundedLevel B (w f)) e (roundedLevel B (w e))} := by |
| rw [roundedWeight_pos_iff hB] |
| exact mem_candidateLabels_iff M w B H e |
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| / |
| theorem candidateLabels_nonloop {n : ℕ} (M : Matroid (Fin n)) |
| (a : Fin n → Option ℤ) (H : Finset (Fin n)) (e : Fin n) |
| (he : e ∈ candidateLabels M a H) : e ∉ M.closure ∅ := by |
| classical |
| exact candidate_not_mem_closure_empty M ⟨H, ∅, ∅, ∅, false⟩ a e (a e) |
| (Finset.mem_filter.mp he).2 |
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| / |
| The assumptions include no positive-rank, nonempty-ground, or no-ties condition. -/ |
| theorem candidate_mass {n : ℕ} (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (w : Weights n) (_hw : ∀ e, 0 ≤ w e) {B : ℝ} (hB : 1 < B) : |
| (∀ H : Finset (Fin n), |
| M.Indep ((positiveGreedy M (fun e => roundedLevel B (w e)) \ H : |
| Finset (Fin n)) : Set (Fin n)) ∧ |
| positiveGreedy M (fun e => roundedLevel B (w e)) \ H ⊆ |
| candidateLabels M (fun e => roundedLevel B (w e)) H) ∧ |
| bitsExpectation (fun _ => (1 / 2 : ℝ)) Finset.univ |
| (fun H => ∑ e ∈ positiveGreedy M (fun e => roundedLevel B (w e)) \ H, |
| roundedWeight B (w e)) = optimum M (fun e => roundedWeight B (w e)) / 2 ∧ |
| bitsExpectation (fun _ => (1 / 2 : ℝ)) Finset.univ |
| (fun H => ∑ e ∈ candidateLabels M (fun e => roundedLevel B (w e)) H, |
| roundedWeight B (w e)) ≤ optimum M (fun e => roundedWeight B (w e)) := by |
| refine ⟨?_, expectation_positiveGreedy_survivors M hE w hB, |
| expectation_candidateLabels_le_optimum M hE w B⟩ |
| intro H |
| exact ⟨positiveGreedy_survivors_indep M hE _ H, |
| positiveGreedy_survivors_subset M _ H⟩ |
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| / |
| theorem optimum_rounding {n : ℕ} (M : Matroid (Fin n)) |
| (w : Weights n) (hw : ∀ e, 0 ≤ w e) {B : ℝ} (hB : 1 < B) : |
| optimum M w / B ≤ optimum M (fun e => roundedWeight B (w e)) ∧ |
| optimum M (fun e => roundedWeight B (w e)) ≤ optimum M w := by |
| constructor |
| · apply (div_le_iff₀ (zero_lt_one.trans hB)).2 |
| simpa only [mul_comm] using optimum_rounding_bound M B hB w hw |
| · exact optimum_mono M _ w (fun e => roundedWeight_le hB (hw e)) |
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| end MatroidProphet.SourceContracts |
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| end OAI |
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