Lower bounds on the lattice-free rank for packing and covering integer programs

TL;DR

This paper establishes lower bounds on the number of cutting plane rounds needed to reach the integer hull in packing and covering integer programs, especially when the integrality gap is large.

Contribution

It provides new lower bounds on the lattice-free rank and split rank, linking them to the integrality gap for packing and covering sets.

Findings

Lower bounds on the split closure rank for packing sets.

Lower bounds on the lattice-free closure rank for packing sets.

Lower bounds on the split rank for covering polyhedra.

Abstract

In this paper, we present lower bounds on the rank of the split closure, the multi-branch closure and the lattice-free closure for packing sets as a function of the integrality gap. We also provide a similar lower bound on the split rank of covering polyhedra. These results indicate that whenever the integrality gap is high, these classes of cutting planes must necessarily be applied for many rounds in order to obtain the integer hull.