The pseudo-Boolean polytope and polynomial-size extended formulations for binary polynomial optimization

TL;DR

This paper introduces the pseudo-Boolean polytope and provides conditions for polynomial-size extended formulations, advancing the understanding of relaxations for binary polynomial optimization problems.

Contribution

It unifies and extends previous results by representing the pseudo-Boolean polytope through signed hypergraphs and establishing conditions for polynomial-size extended formulations.

Findings

Polytope representation via signed hypergraphs

Sufficient conditions for polynomial-size extended formulations

Unification of prior results on binary polynomial optimization

Abstract

With the goal of obtaining strong relaxations for binary polynomial optimization problems, we introduce the pseudo-Boolean polytope defined as the convex hull of the set of binary points satisfying a collection of equations containing pseudo-Boolean functions. By representing the pseudo-Boolean polytope via a signed hypergraph, we obtain sufficient conditions under which this polytope has a polynomial-size extended formulation. Our new framework unifies and extends all prior results on the existence of polynomial-size extended formulations for the convex hull of the feasible region of binary polynomial optimization problems of degree at least three.