Mixed integer reformulations of integer programs and the affine TU-dimension of a matrix
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.
KeywordsInteger programming Master knapsack problem Total unimodularity
Mathematics Subject Classification90C10 Integer programming 90C11 Mixed integer programming
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.
- 2.Baum, S., Trotter, L.E.: Optimization and operations research. In: Proceedings of a Workshop Held at the University of Bonn, October 2–8, 1977, chapter Integer rounding and polyhedral decomposition for totally unimodular systems, pp. 15–23. Springer Berlin Heidelberg, (1978)Google Scholar
- 3.Carr, R.D., Konjevod, G.: Polyhedral combinatorics. In: Greenberg, H. (ed.) Tutorials on Emerging Methodologies and Applications in Operations Research, pp. 1–48. Springer, Berlin (2004)Google Scholar
- 15.Lodi, A.: Personal communication, 2014 and 2015Google Scholar
- 22.Vanderbeck, F., Wolsey, L.A.: Reformulation and decomposition of integer programs. In: Jünger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds.) 50 Years of Integer Programming 1958–2008, pp. 431–502. Springer, Berlin (2010)Google Scholar
- 23.Veselov, S.I., Gribanov, D.V.: On integer programming with almost unimodular matrices and the flatness theorem for simplices (2015). arXiv:1505.03132 [cs.CG]