Two-halfspace closure

TL;DR

This paper introduces the two-halfspace closure, a new polyhedral closure for integer programs that generalizes the Chvátal-Gomory closure and relates to split ranks, with implications for understanding polyhedral structures.

Contribution

It defines the two-halfspace closure, proves its polyhedrality, and analyzes its relationship with split rank, providing new insights into polyhedral closures in integer programming.

Findings

The two-halfspace closure is polyhedral.

The two-halfspace rank is at most the split rank.

Split rank can be up to twice the two-halfspace rank.

Abstract

We define a new cutting plane closure for pure integer programs called the two-halfspace closure. It is a natural generalization of the well-known Chv\'atal-Gomory closure. We prove that the two-halfspace closure is polyhedral. We also study the corresponding $2$-halfpsace rank of any valid inequality and show that it is at most the split rank of the inequality. Moreover, while the split rank can be strictly larger than the two-halfspace rank, the split rank is at most twice the two-halfspace rank. A key step of our analysis shows that the split closure of a rational polyhedron can be obtained by considering the split closures of all $k$-dimensional (rational) projections of the polyhedron, for any fixed $k\geq 2$. This result may be of independent interest.