Simple odd $\beta$-cycle inequalities for binary polynomial optimization

TL;DR

This paper introduces simple odd β-cycle inequalities for binary polynomial optimization, providing a polynomial-time separation algorithm and demonstrating their effectiveness in closing integrality gaps through computational experiments.

Contribution

The paper proposes a weaker, computationally efficient version of odd β-cycle inequalities with a polynomial-time separation algorithm, extending their applicability to arbitrary instances.

Findings

The inequalities have Chvátal rank 2.

They help close the integrality gap in experiments.

The implementation shows room for computational improvements.

Abstract

We consider the multilinear polytope which arises naturally in binary polynomial optimization. Del Pia and Di Gregorio introduced the class of odd $\beta$-cycle inequalities valid for this polytope, showed that these generally have Chv{\'a}tal rank 2 with respect to the standard relaxation and that, together with flower inequalities, they yield a perfect formulation for cycle hypergraph instances. Moreover, they describe a separation algorithm in case the instance is a cycle hypergraph. We introduce a weaker version, called simple odd $\beta$-cycle inequalities, for which we establish a strongly polynomial-time separation algorithm for arbitrary instances. These inequalities still have Chv{\'a}tal rank 2 in general and still suffice to describe the multilinear polytope for cycle hypergraphs. Finally, we report about computational results of our prototype implementation. The simple odd $\beta$-cycle inequalities sometimes help to close more of the integrality gap in the experiments; however, the preliminary implementation has substantial computational cost, suggesting room for improvement in the separation algorithm.