Maximal lattice free bodies, test sets and the Frobenius problem

TL;DR

This paper presents an efficient algorithm for computing maximal lattice free bodies related to a fixed matrix, generalizing the Frobenius problem to higher dimensions using algebraic and combinatorial methods.

Contribution

It introduces a new algorithm for maximal lattice free bodies and extends the Frobenius problem solution to arbitrary dimensions using algebraic techniques.

Findings

Algorithm efficiently computes maximal lattice free bodies.

Generalization of the Frobenius problem to higher dimensions.

Utilizes algebraic methods like Groebner bases for computation.

Abstract

Maximal lattice free bodies are maximal polytopes without interior integral points. Scarf initiated the study of maximal lattice free bodies relative to the facet normals in a fixed matrix. In this paper we give an efficient algorithm for computing the maximal lattice free bodies of an integral matrix A. An important ingredient is a test set for a certain integer program associated with A. This test set may be computed using algebraic methods. As an application we generalize the Scarf-Shallcross algorithm for the three-dimensional Frobenius problem to arbitrary dimension. In this context our method is inspired by the novel algorithm by Einstein, Lichtblau, Strzebonski and Wagon and the Groebner basis approach by Roune.