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

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 $S$, 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 that arise in combinatorial optimization. We propose a new efficient method that interpolates between these two approaches. Our procedure strengthens any convex set $Q \subseteq \mathbb{R}ⁿ$ 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 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.