Characterizing Polytopes Contained in the $0/1$-Cube with Bounded Chvátal-Gomory Rank

Benchetrit, Yohann; Fiorini, Samuel; Huynh, Tony; Weltge, Stefan

doi:10.48550/arxiv.1611.06593

Cited by 1 publication

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“…2 To avoid possible confusion, we warn the reader that in a previous version of [6], this notion is called pitch instead of notch. 3 Here, this means that min{c ⊺ x :…”

Section: Resultsmentioning

confidence: 99%

“…We extend the definition of notch from inequalities to sets of 0/1-points as follows. The notch of a non-empty set S ⊆ {0, 1} n , denoted ν(S), is the largest notch of any inequality in standard form that is valid for S. It can be shown that ν(S) is equal to the smallest number k such that every k-dimensional face of [0, 1] n contains a point from S. This equivalent definition of notch 2 was introduced in [6]. The main result of [6] is that if S has bounded notch and conv(S) has bounded facet coefficients, then every polytope Q ⊆ [0, 1] n whose set of 0/1-points is S has bounded Chvátal-Gomory rank.…”

Section: Measuring the Strength: Pitch And Notchmentioning

confidence: 99%

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Strengthening Convex Relaxations of 0/1-Sets Using Boolean Formulas

Fiorini¹,

Huynh²,

Weltge³

2017

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In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set S of feasible integer points, such as popular linear programming or semi-definite programming hierarchies. On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets S that arise in combinatorial optimization.We propose a new efficient method that interpolates between these two approaches. Our procedure strengthens any convex set containing a set S ⊆ {0, 1} n by exploiting certain additional information about S. Namely, the required extra information will be in the form of a Boolean formula defining the target set S.The aim of this work is to analyze various aspects regarding the strength of our procedure. As one result, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems.

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“…2 To avoid possible confusion, we warn the reader that in a previous version of [6], this notion is called pitch instead of notch. 3 Here, this means that min{c ⊺ x :…”

Section: Resultsmentioning

confidence: 99%

Section: Measuring the Strength: Pitch And Notchmentioning

confidence: 99%