N-fold integer programming

Authors:
JesúS A. De Loera;Raymond Hemmecke;Shmuel Onn;Robert Weismantel
Affiliations:
University of California at Davis, Davis, CA 95616, USA;Otto-von-Guericke Universität Magdeburg, D-39106 Magdeburg, Germany;Technion - Israel Institute of Technology, 32000 Haifa, Israel;Otto-von-Guericke Universität Magdeburg, D-39106 Magdeburg, Germany
Venue:
Discrete Optimization
Year:
2008

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Abstract

In this article we study a broad class of integer programming problems in variable dimension. We show that these so-termed n-fold integer programming problems are polynomial time solvable. Our proof involves two heavy ingredients discovered recently: the equivalence of linear optimization and the so-called directed augmentation, and the stabilization of certain Graver bases. We discuss several applications of our algorithm to multiway transportation problems and to packing problems. One important consequence of our results is a polynomial time algorithm for the d-dimensional integer transportation problem for long multiway tables. Another interesting application is a new algorithm for the classical cutting-stock problem.