Abstract
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 subseteq {0,1}^n by exploiting certain additional information about S. Namely, the required extra information will be in the form of a Boolean formula phi defining the target set S. The new relaxation is obtained by “feeding” the convex set into the formula phi . 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.
Highlights
In convex integer programming, there exist various procedures to strengthen convex relaxations of sets of integer points
We demonstrate how our findings together with Rothvoß’ result [25] on the extension complexity of the matching polytope yield a very simple proof of a seminal result of Raz and Wigderson [23, Theorem 4.1], which states4 that any monotone Boolean formula deciding whether a graph on n nodes contains a perfect matching has size 2 (n)
There is a polyhedral relaxation P of S such that all points of P satisfy all valid inequalities of pitch at most k, and P can be defined by an extended formulation of size at most 2n ·k
Summary
There exist various procedures to strengthen convex relaxations of sets of integer points. It was recently shown by Rothvoß that all extended formulations of the matching polytope have exponential size [25] Another property of our procedure is that it is complete in the sense that iterating it a finite number of times (at most n times) always yields the convex hull of S. Despite the simplicity of our method, we obtain extended formulations whose size is often vastly smaller than those of Bienstock and Zuckerberg [7,8] and Mastrolilli [22]. We discuss this in more detail in Sect.
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