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Mathematical Programming

, Volume 169, Issue 2, pp 565–584 | Cite as

Mixed integer reformulations of integer programs and the affine TU-dimension of a matrix

  • Jörg Bader
  • Robert Hildebrand
  • Robert Weismantel
  • Rico Zenklusen
Full Length Paper Series A

Abstract

We study the reformulation of integer linear programs by means of a mixed integer linear program with fewer integer variables. Such reformulations can be solved efficiently with mixed integer linear programming techniques. We exhibit examples that demonstrate how integer programs can be reformulated using far fewer integer variables. To this end, we introduce a generalization of total unimodularity called the affine TU-dimension of a matrix and study related theory and algorithms for determining the affine TU-dimension of a matrix. We also present bounds on the number of integer variables needed to represent certain integer hulls.

Keywords

Integer programming Master knapsack problem Total unimodularity 

Mathematics Subject Classification

90C10 Integer programming 90C11 Mixed integer programming 

Notes

Acknowledgements

We thank Santanu S. Dey for discussing his idea for the lower bound in Example 8. We owe thanks to Shmuel Onn who made us aware of a much simplified version of the proof of Theorem 18. We also want to express our gratitude to two anonymous reviewers. Their detailed comments and suggestions on an earlier version of the manuscript led to enhancements on the general structure of our paper, as well as greatly improved the paper in many ways.

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Institute for Operations ResearchETH ZürichZürichSwitzerland
  2. 2.IBM T.J. Watson Research CenterYorktown HeightsUSA

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