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Valid separating inequalities from independent-set bounds

Lax263307.ComponentCuts · concepts/Lax263307/ComponentCuts.lean · lax-263307

proven

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    Natural Language Statement

    Theorem

    Suppose every independent set in G[C]G[C] has size at most uu. For ∣A∣≤r|A|\le r, every z∈Pr(G,k)z\in P_r(G,k) satisfies

    ∑v∈CzA∪{v}≤uzA.\sum_{v\in C}z_{A\cup\{v\}}\le u z_A.

    A query yy with yA>0y_A>0 violates this inequality precisely when its conditioned costs satisfy ∑v∈Cxv>u\sum_{v\in C}x_v>u.

    Concept map
    2 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 valid_cut proven

    2 violation_iff proven

    Lean source view on GitHub

    1import Lax263307.LinearRelaxation
    2
    3/-!
    4---
    5title: Valid separating inequalities from independent-set bounds
    6type: theorem
    7---
    8Suppose every independent set in G[C]G[C] has size at most uu.
    9For ∣A∣≤r|A|\le r, every z∈Pr(G,k)z\in P_r(G,k) satisfies
    10∑v∈CzA∪{v}≤uzA.\sum_{v\in C}z_{A\cup\{v\}}\le u z_A.
    11A query yy with yA>0y_A>0 violates this inequality precisely when its
    12conditioned costs satisfy ∑v∈Cxv>u\sum_{v\in C}x_v>u.
    13
    14# Attribution
    15
    16Bonnet–Chang, Lemma 13. The set CC need not be a connected component;
    17the algorithm applies this statement to the components left by a separator.
    18-/
    19
    20namespace Lax263307.ComponentCuts
    21
    22open LinearRelaxation
    23
    24/-- A certified independence bound supplies an inequality valid on the exact hull. -/
    25axiom valid_cut {n : ℕ} (G : SimpleGraph (Fin n)) (k r : ℕ)
    26 (C : Finset (Fin n)) (u : ℕ) (bound : IsIndependenceBound G C u)
    27 (A : Finset (Fin n)) (size : A.card ≤ r)
    28 (z : MomentVector n r) (member : z ∈ independentSetHull G k r) :
    29 ∑ v ∈ C, value z (insert v A) ≤ (u : ℝ) * value z A
    30
    31/-- The exact rational test used to detect a violated cut. -/
    32axiom violation_iff {n r : ℕ} (y : MomentVector n r)
    33 (A C : Finset (Fin n)) (u : ℕ) (positive : 0 < value y A) :
    34 (u : ℝ) < ∑ v ∈ C, conditioned y A v ↔
    35 (u : ℝ) * value y A < ∑ v ∈ C, value y (insert v A)
    36
    37end Lax263307.ComponentCuts
    38
    Show ProofShow Proof

    Attribution

    Bonnet–Chang, Lemma 13. The set CC need not be a connected component; the algorithm applies this statement to the components left by a separator.

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