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arXiv · 1010.1077

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

Gennadiy Averkov·Christian Wagner·Robert Weismantel
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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.

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Gennadiy Averkov, Christian Wagner, Robert Weismantel. 2010-10-06. Maximal lattice-free polyhedra: finiteness and an explicit description in dimension three. https://arxiv.org/abs/1010.1077

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