Mathematical Programming

, Volume 154, Issue 1–2, pp 407–425 | Cite as

Lower bounds on the sizes of integer programs without additional variables

Full Length Paper Series B

Abstract

Let \( X \) be the set of integer points in some polyhedron. We investigate the smallest number of facets of any polyhedron whose set of integer points is \( X \). This quantity, which we call the relaxation complexity of \( X \), corresponds to the smallest number of linear inequalities of any integer program having \( X \) as the set of feasible solutions that does not use auxiliary variables. We show that the use of auxiliary variables is essential for constructing polynomial size integer programming formulations in many relevant cases. In particular, we provide asymptotically tight exponential lower bounds on the relaxation complexity of the integer points of several well-known combinatorial polytopes, including the traveling salesman polytope and the spanning tree polytope. In addition to the material in the extended abstract Kaibel and Weltge (2014) we include omitted proofs, supporting figures, discussions about properties of coefficients in such formulations, and facts about the complexity of formulations in more general settings.

Keywords

Integer programming Relaxations Auxiliary variables TSP 

Mathematics Subject Classification

52Bxx 

Notes

Acknowledgments

We would like to thank Gennadiy Averkov for valuable comments on this work.

References

  1. 1.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  2. 2.
    Avis, D., Tiwary, H.R.: On the extension complexity of combinatorial polytopes. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M.Z., Peleg, D. (eds.) Automata, Languages, and Programming, Lecture Notes in Computer Science, vol. 7965, pp. 57–68. Springer, Berlin, Heidelberg (2013). doi:10.1007/978-3-642-39206-1_6
  3. 3.
    Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., de Wolf, R.: Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds. In: STOC, pp. 95–106 (2012)Google Scholar
  4. 4.
    Gavish, B., Graves, S.C.: The travelling salesman problem and related problems. Tech. rep, Operations Research Center, Massachusetts Institute of Technology (1978)Google Scholar
  5. 5.
    Goemans, M.: Smallest compact formulation for the permutahedron (to appear in a forthcoming issue of Mathematical Programming, Series B “Lifts of Convex Sets” edited by V. Kaibel and R. Thomas) (2014). http://www-math.mit.edu/~goemans/publ.html
  6. 6.
    Jeroslow, R.: On defining sets of vertices of the hypercube by linear inequalities. Discrete Math. 11(2), 119–124 (1975)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Kaibel, V., Weltge, S.: Lower bounds on the sizes of integer programs without additional variables. In: J. Lee, J. Vygen (eds.) Integer Programming and Combinatorial Optimization. Proceedings of IPCO XVII, Bonn, Lecture Notes in Computer Science, vol. 8494. Springer, Berlin (2014)Google Scholar
  8. 8.
    Lee, J., Margot, F.: On a binary-encoded ilp coloring formulation. INFORMS J. Comput. 19(3), 406–415 (2007)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Martin, R.K.: Using separation algorithms to generate mixed integer model reformulations. Oper. Res. Lett. 10(3), 119–128 (1991). http://www.sciencedirect.com/science/article/pii/016763779190028N
  10. 10.
    Miller, C.E., Tucker, A.W., Zemlin, R.A.: Integer programming formulation of traveling salesman problems. J. ACM 7(4), 326–329 (1960)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Minkowski, H.: Geometrie der Zahlen. Teubner Verlag, Leipzig (1896)Google Scholar
  12. 12.
    Pokutta, S., Vyve, M.V.: A note on the extension complexity of the knapsack polytope. Oper. Res. Lett. 41(4), 347–350 (2013). doi:10.1016/j.orl.2013.03.010. http://www.sciencedirect.com/science/article/pii/S0167637713000394
  13. 13.
    Rado, R.: An inequality. J. Lond. Math. Soc. 27, 1–6 (1952)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Rothvoss, T.: The matching polytope has exponential extension complexity. In: Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC ’14. ACM, New York, NY, pp. 263–272 (2014). doi:10.1145/2591796.2591834
  15. 15.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York, NY (1986)MATHGoogle Scholar
  16. 16.
    Schrijver, A.: Combinatorial Optimization—Polyhedra and Efficiency. Springer, Berlin (2003)MATHGoogle Scholar
  17. 17.
    Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  1. 1.Otto-von-Guericke-Universität MagdeburgMagdeburgGermany

Personalised recommendations