Lifting properties of maximal lattice-free polyhedra

TL;DR

This paper investigates the properties of minimal liftings of cut-generating functions derived from maximal lattice-free polyhedra, establishing invariance, constructing new examples, and extending previous characterizations to broader classes.

Contribution

It generalizes existing results on lifting properties from simplicial to arbitrary maximal lattice-free polyhedra and introduces a versatile construction method for such polyhedra with unique-lifting properties.

Findings

Proves invariance of unique minimal liftings for general maximal lattice-free polyhedra.

Develops a general iterative construction for maximal lattice-free polyhedra with the unique-lifting property.

Extends characterizations of lifting from simplices to more general polytopes using discrete geometry results.

Abstract

We study the uniqueness of minimal liftings of cut-generating functions obtained from maximal lattice-free polyhedra. We prove a basic invariance property of unique minimal liftings for general maximal lattice-free polyhedra. This generalizes a previous result by Basu, Cornu\'ejols and K\"oppe~\cite{bcm} for {\em simplicial} maximal lattice-free polytopes, thus completely settling this fundamental question about lifting for maximal lattice-free polyhedra. We further give a very general iterative construction to get maximal lattice-free polyhedra with the unique-lifting property in arbitrary dimensions. This single construction not only obtains all previously known polyhedra with the unique-lifting property, but goes further and vastly expands the known list of such polyhedra. Finally, we extend characterizations from~\cite{bcm} about lifting with respect to maximal lattice-free simplices to more general polytopes. These nontrivial generalizations rely on a number of results from discrete geometry, including the Venkov-Alexandrov-McMullen theorem on translative tilings and characterizations of zonotopes in terms of central symmetry of their faces.