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

TL;DR

This paper introduces a new method to strengthen convex relaxations of 0/1 sets in integer programming by leveraging Boolean formulas, bridging general-purpose and set-specific approaches, and analyzing its effectiveness.

Contribution

It proposes an efficient interpolation method that uses Boolean formulas to enhance convex relaxations of 0/1 sets, extending previous hierarchy results.

Findings

Simplifies existing hierarchies for covering problems

Improves bounds on relaxation strength

Extends analysis of Boolean formula-based strengthening

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.