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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 79 80 81 82 83 84 85 86 87 88 89 | import OAI.Probability.MatroidSecretary.Selection.CandidateSpecification
import OAI.Probability.MatroidSecretary.Selection.CandidateContracts
namespace OAI
/-! Correspondence of the independent source definitions with the implemented
candidate filter and positive priority-greedy, followed by the source contract. -/
namespace MatroidProphet.CandidateSpecification
open Finset
lemma higher_eq_main {n : ℕ} (f : Fin n) (a : Option ℤ)
(e : Fin n) (b : Option ℤ) :
higher f a e b ↔ MainAlgorithm.higher f a e b := Iff.rfl
lemma candidates_eq_main {n : ℕ} (M : Matroid (Fin n))
(a : Fin n → Option ℤ) (H : Finset (Fin n)) :
candidates M a H = MainAlgorithm.candidateLabels M a H := by
classical
rw [MainAlgorithm.candidateLabels, MainAlgorithm.candidateSet_eq_filter]
ext element
by_cases hpositive : a element = none
· simp [candidates, hpositive]
· have hprior : {label | label ∈ H ∧ higher label (a label) element (a element)} =
Candidates.prior (Priority.time a) H element := by
ext label
simp only [Candidates.prior, Set.mem_ofPred_eq, higher_eq_main,
Priority.main_higher_iff_time a label element hpositive]
simp [candidates, Candidates.candidates, hpositive, hprior]
lemma greedy_eq_main {n : ℕ} (M : Matroid (Fin n))
(a : Fin n → Option ℤ) : greedy M a = MainAlgorithm.positiveGreedy M a := by
classical
ext e
by_cases ha : a e = none
· simp [greedy, MainAlgorithm.positiveGreedy, ha]
· have hp : {f | higher f (a f) e (a e)} =
Candidates.prior (Priority.time a) Finset.univ e := by
ext f
simp only [Candidates.prior, Finset.mem_univ, true_and,
Set.mem_ofPred_eq, higher_eq_main, Priority.main_higher_iff_time a f e ha]
simp [greedy, MainAlgorithm.positiveGreedy, Candidates.greedy, ha, hp]
lemma survivors_eq_main {n : ℕ} (M : Matroid (Fin n))
(a : Fin n → Option ℤ) (H : Finset (Fin n)) :
survivors M a H = MainAlgorithm.positiveGreedy M a \ H := by
rw [survivors, greedy_eq_main]
theorem candidateMass : CandidateMass := by
intro n M hE w
simpa only [survivors_eq_main, candidates_eq_main] using
CandidateContracts.candidate_mass M hE w
/-- Candidate membership uses only the focal key and keys on the revealed mask,
never another unobserved label's key. -/
lemma candidate_membership_congr {n : ℕ} (M : Matroid (Fin n))
(a a' : Fin n → Option ℤ) (H : Finset (Fin n)) (e : Fin n)
(he : a e = a' e) (hH : ∀ f ∈ H, a f = a' f) :
e ∈ candidates M a H ↔ e ∈ candidates M a' H := by
classical
have hp : {f | f ∈ H ∧ higher f (a f) e (a e)} =
{f | f ∈ H ∧ higher f (a' f) e (a' e)} := by
ext f
by_cases hf : f ∈ H
· simp [hf, he, hH f hf]
· simp [hf]
simp only [candidates, Finset.mem_filter, Finset.mem_univ, true_and]
rw [hp, he]
/-- The source greedy definition really attains rounded OPT, with the source's
rounding comparison on both sides. This verifies the ancillary input `I*` rather
than assuming its optimality. -/
theorem source_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 := greedy 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
simpa only [greedy_eq_main] using CandidateContracts.rounding_and_greedy M hE w hw
end MatroidProphet.CandidateSpecification
end OAI
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