Strengthening convex relaxations of 0/1-sets using Boolean formulas

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 popul...

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Bibliographic Details
Published in:Mathematical programming Vol. 190; no. 1-2; pp. 467 - 482
Main Authors: Fiorini, Samuel, Huynh, Tony, Weltge, Stefan
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2021
Springer Nature B.V
Subjects:
Boolean
Boolean algebra
Calculus of Variations and Optimal Control; Optimization
Combinatorial analysis
Combinatorics
Full Length Paper
Hierarchies
Integer programming
Linear programming
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Theoretical
90Cxx
68Q06
52Bxx
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary: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 ϕ defining the target set S . The new relaxation is obtained by “feeding” the convex set into the formula ϕ . We analyze various aspects regarding the strength of our procedure. As one application, 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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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-020-01542-w