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

Abstract

AbstractIn 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 $$ S \subseteq \{0,1\}^n $$ 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 $$\phi $$ ϕ defining the target set S. The new relaxation is obtained by “feeding” the convex set into the formula $$\phi $$ 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.