Notions of maximality for integral lattice-free polyhedra: the case of dimension three

TL;DR

This paper proves that in three dimensions, the concepts of $ ext{maximality}$ for integral lattice-free polyhedra with respect to integer points and real points are equivalent, simplifying classification efforts in this case.

Contribution

The paper establishes the equivalence of $ ext{Z}^3$-maximality and $ ext{R}^3$-maximality for integral polyhedra, completing the understanding for three-dimensional cases.

Findings

Proves equivalence of $ ext{Z}^3$-maximality and $ ext{R}^3$-maximality in 3D.

Shows classification of $ ext{R}^3$-maximal polyhedra includes all $ ext{Z}^3$-maximal ones.

Completes the dimension-specific analysis of lattice-free polyhedra.

Abstract

Lattice-free sets (convex subsets of $\mathbb{R}^d$ without interior integer points) and their applications for cutting-plane methods in mixed-integer optimization have been studied in recent literature. Notably, the family of all integral lattice-free polyhedra which are not properly contained in another integral lattice-free polyhedron has been of particular interest. We call these polyhedra $\mathbb{Z}^d$-maximal. It is known that, for fixed $d$, the family $\mathbb{Z}^d$-maximal integral lattice-free polyhedra is finite up to unimodular equivalence. In view of possible applications in cutting-plane theory, one would like to have a classification of this family. However, this turns out to be a challenging task already for small dimensions. In contrast, the subfamily of all integral lattice-free polyhedra which are not properly contained in any other lattice-free set, which we call $\mathbb{R}^d$-maximal lattice-free polyhedra, allow a rather simple geometric characterization. Hence, the question was raised for which dimensions the notions of $\mathbb{Z}^d$-maximality and $\mathbb{R}^d$-maximality are equivalent. This was known to be the case for dimensions one and two. On the other hand, Nill and Ziegler (2011) showed that for dimension $d \ge 4$, there exist polyhedra which are $\mathbb{Z}^d$-maximal but not $\mathbb{R}^d$-maximal. In this article, we consider the remaining case $d = 3$ and prove that for integral polyhedra the notions of $\mathbb{R}^3$-maximality and $\mathbb{Z}^3$-maximality are equivalent. As a consequence, the classification of all $\mathbb{R}^3$-maximal integral polyhedra by Averkov, Wagner and Weismantel (2011) contains all $\mathbb{Z}^3$-maximal integral polyhedra.