Approximation of corner polyhedra with families of intersection cuts

TL;DR

This paper characterizes the complexity of lattice-free sets needed to approximate corner polyhedra within a constant factor, revealing a threshold based on the number of facets relative to the problem dimension.

Contribution

It provides a precise characterization of the facet complexity required for constant factor approximations of corner polyhedra, depending on the problem dimension.

Findings

Approximation up to a constant factor is possible with lattice-free sets having more than 2^{n-1} facets.

No such approximation is possible with sets having at most 2^{n-1} facets.

When approximation depends on the fractional vertex denominator, the threshold reduces to n facets.

Abstract

We study the problem of approximating the corner polyhedron using intersection cuts derived from families of lattice-free sets in $\mathbb{R}^n$. In particular, we look at the problem of characterizing families that approximate the corner polyhedron up to a constant factor, which depends only on $n$ and not the data or dimension of the corner polyhedron. The literature already contains several results in this direction. In this paper, we use the maximum number of facets of lattice-free sets in a family as a measure of its complexity and precisely characterize the level of complexity of a family required for constant factor approximations. As one of the main results, we show that, for each natural number $n$, a corner polyhedron with $n$ basic integer variables and an arbitrary number of continuous non-basic variables is approximated up to a constant factor by intersection cuts from lattice-free sets with at most $i$ facets if $i> 2^{n-1}$ and that no such approximation is possible if $i \leq 2^{n-1}$. When the approximation factor is allowed to depend on the denominator of the fractional vertex of the linear relaxation of the corner polyhedron, we show that the threshold is $i > n$ versus $i \leq n$. The tools introduced for proving such results are of independent interest for studying intersection cuts.