Maximal lattice free bodies, test sets and the Frobenius problem

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 matr...

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Bibliographic Details
Published inarXiv.org
Main Authors Jensen, Anders, Lauritzen, Niels, Roune, Bjarke
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 30.05.2007
Subjects
Algorithms
Integrals
Polytopes
Rings (mathematics)
Test sets
Wagons
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Summary: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.
ISSN:2331-8422