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


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.


Integer programming Master knapsack problem Total unimodularity 

Mathematics Subject Classification

90C10 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.


  1. 1.
    Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. Discrete Appl. Math. 89(1–3), 3–44 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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. 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
  4. 4.
    Conforti, M., Cornuéjols, G., Vušković, K.: Balanced matrices. Discrete Math. 306(19–20), 2411–2437 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Eggan, L., Plantholt, M.: The chromatic index of nearly bipartite multigraphs. J. Comb. Theory Ser. B 40(1), 71–80 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gijswijt, D.: Integer decomposition for polyhedra defined by nearly totally unimodular matrices. SIAM J. Discrete Math. 19(3), 798–806 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hassin, R., Levin, A.: An efficient polynomial time approximation scheme for the constrained minimum spanning tree problem using matroid intersection. SIAM J. Comput. 33(2), 261–268 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Heller, I.: On linear systems with integral valued solutions. Pac. J. Math. 7(3), 1351–1364 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hibi, T., Higashitani, A., Katthän, L., Okazaki, R.: Normal cyclic polytopes and cyclic polytopes that are not very ample. J. Aust. Math. Soc. 96, 61–77 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jeroslow, R.: On defining sets of vertices of the hypercube by linear inequalities. Discrete Math. 11(2), 119–124 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kaibel, V., Pashkovich, K.: Constructing extended formulations from reflection relations. In: Jünger, M., Reinelt, G. (eds.) Facets of Combinatorial Optimization - Festschrift for Martin Grötschel, pp. 77–100. Springer, Berlin Heidelberg (2013)CrossRefGoogle Scholar
  12. 12.
    Karzanov, A.V., McCormick, S.T.: Polynomial methods for separable convex optimization in unimodular linear spaces with applications. SIAM J. Comput. 26(4), 1245–1275 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms, 4th edn. Springer, Berlin (2007)zbMATHGoogle Scholar
  14. 14.
    Lenstra Jr., H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lodi, A.: Personal communication, 2014 and 2015Google Scholar
  16. 16.
    Martin, R.K.: Generating alternative mixed-integer programming models using variable redefinition. Oper. Res. 35(6), 820–831 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Oriolo, G., Sanità, L., Zenklusen, R.: Network design with a discrete set of traffic matrices. Oper. Res. Lett. 41(4), 390–396 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Padberg, M.: Total unimodularity and the Euler-subgraph problem. Oper. Res. Lett. 7(4), 173–179 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Schrijver, A.: Theory of Linear and Integer Programming. John Wiley and Sons, New York (1986)zbMATHGoogle Scholar
  20. 20.
    Seymour, P.D.: Decomposition of regular matroids. J. Comb. Theory Ser. B 28(3), 305–359 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Truemper, K.: A decomposition theory for matroids. V. Testing of matrix total unimodularity. J. Comb. Theory Ser. B 49(2), 241–281 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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. 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]
  24. 24.
    Woeginger, G.J., Yu, Z.: On the equal-subset-sum problem. Inf. Process. Lett. 42(6), 299–302 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

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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