TY - UNPB

T1 - Maximal lattice free bodies, test sets and the Frobenius problem

AU - Jensen, Anders Nedergaard

AU - Lauritzen, Niels

AU - Roune, Bjarke Hammersholt

N1 - Cite as: arXiv:0705.4439v2 [math.CO]

PY - 2007

Y1 - 2007

N2 - 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.

AB - 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.

M3 - Working paper

BT - Maximal lattice free bodies, test sets and the Frobenius problem

PB - arxiv.org

ER -