Maximal lattice-free polyhedra: finiteness and an explicit description in dimension three

TL;DR

This paper proves the finiteness of maximal lattice-free rational polyhedra of a given precision in any dimension and provides a complete classification of all such polyhedra in three dimensions, advancing cutting plane theory.

Contribution

It establishes the finiteness of maximal lattice-free rational polyhedra up to affine transformations and explicitly classifies all maximal lattice-free integral polyhedra in three dimensions.

Findings

Finiteness of maximal lattice-free rational polyhedra for fixed precision

Complete list of maximal lattice-free integral polyhedra in dimension three

Results support cutting plane methods in mixed-integer linear optimization

Abstract

A convex set with nonempty interior is maximal lattice-free if it is inclusion-maximal with respect to the property of not containing integer points in its interior. Maximal lattice-free convex sets are known to be polyhedra. The precision of a rational polyhedron $P$ in $\mathbb{R}^d$ is the smallest integer $s$ such that $sP$ is an integral polyhedron. In this paper we show that, up to affine mappings preserving $\mathbb{Z}^d$, the number of maximal lattice-free rational polyhedra of a given precision $s$ is finite. Furthermore, we present the complete list of all maximal lattice-free integral polyhedra in dimension three. Our results are motivated by recent research on cutting plane theory in mixed-integer linear optimization.