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

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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.

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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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Authors and Affiliations

  1. Institute for Operations Research, ETH Zürich, Zürich, Switzerland

    Jörg Bader, Robert Weismantel & Rico Zenklusen

  2. IBM T.J. Watson Research Center, Yorktown Heights, NY, USA

    Robert Hildebrand

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  1. Jörg Bader
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  2. Robert Hildebrand
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  3. Robert Weismantel
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  4. Rico Zenklusen
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Correspondence to Robert Hildebrand.

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Bader, J., Hildebrand, R., Weismantel, R. et al. Mixed integer reformulations of integer programs and the affine TU-dimension of a matrix. Math. Program. 169, 565–584 (2018). https://doi.org/10.1007/s10107-017-1147-2

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  • DOI: https://doi.org/10.1007/s10107-017-1147-2

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