File size: 3,584 Bytes
a2ee19a | 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 | 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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