Advertisement

Mathematical Programming

, Volume 154, Issue 1–2, pp 381–406 | Cite as

Simple extensions of polytopes

  • Volker Kaibel
  • Matthias Walter
Full Length Paper Series B
  • 245 Downloads

Abstract

We introduce the simple extension complexity of a polytope \(P\) as the smallest number of facets of any simple (i.e., non-degenerate in the sense of linear programming) polytope which can be projected onto \(P\). We devise a combinatorial method to establish lower bounds on the simple extension complexity and show for several polytopes that they have large simple extension complexities. These examples include both the spanning tree and the perfect matching polytopes of complete graphs, uncapacitated flow polytopes for non-trivially decomposable directed acyclic graphs, hypersimplices, and random 0/1-polytopes with vertex numbers within a certain range. On our way to obtain the result on perfect matching polytopes we generalize a result of Padberg and Rao’s on the adjacency structures of those polytopes. In addition to the material in the extended abstract (Kaibel and Walter in Integer programming and combinatorial optimization. Lecture Notes in Computer Science, vol 8494. Springer, Berlin, 2014) we include omitted proofs, supporting figures, and an analysis of known upper bounding techniques.

Keywords

Extended formulations Simple polytopes Matching polytopes  Flow polytopes Spanning tree polytopes 

Mathematics Subject Classification

52B99 

Notes

Acknowledgments

We are greatful to the referees whose comments lead to significant improvements in the presentation of the material.

References

  1. 1.
    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, volume 7965 of Lecture Notes in Computer Science, pp. 57–68. Springer, Berlin (2013)Google Scholar
  2. 2.
    Balas, E.: Disjunctive Programming: Properties of the Convex Hull of Feasible Points. MSRR 348, Carnegie Mellon University, Pittsburg, PA (1974)Google Scholar
  3. 3.
    Balas, E.: Disjunctive programming. In: Johnson, E.L., Hammer, P.L., Korte, B.H. (eds.) Discrete Optimization II. Annals of Discrete Mathematics, vol. 5, pp. 3–51. Elsevier (1979). http://dx.doi.org/10.1016/S0167-5060(08)70342-X
  4. 4.
    Bienstock, D.: Approximate formulations for 0–1 knapsack sets. Oper. Res. Lett. 36(3), 317–320 (2008)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Carr, R.D., Konjevod, G.: Polyhedral combinatorics. In: Greenberg, H.J. (ed.) Tutorials on Emerging Methodologies and Applications in Operations Research volume 76 of International Series in Operations Research and Management Science, chapter 2, pp. 1–46. Springer, Berlin (2005)Google Scholar
  6. 6.
    Chvátal, V.: On certain polytopes associated with graphs. J. Combin. Theory Ser. B 18(2), 138–154 (1975)MATHCrossRefGoogle Scholar
  7. 7.
    Dantzig, G.B.: Linear Programming and Extensions. Princeton Landmarks in Mathematics and Physics. Princeton University Press, Princeton (1963)Google Scholar
  8. 8.
    Fiorini, S., Kaibel, V., Pashkovich, K., Theis, D.O.: Combinatorial bounds on nonnegative rank and extended formulations. Discrete Math. 313(1), 67–83 (2013)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., de Wolf, R.: Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds. In: Karloff, H.J., Pitassi, T. (eds.) STOC, pp. 95–106. ACM, New York (2012)Google Scholar
  10. 10.
    Gallo, G., Sodini, C.: Extreme points and adjacency relationship in the flow polytope. Calcolo 15, 277–288 (1978). doi: 10.1007/BF02575918 MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Gillmann, R.: 0/1-Polytopes typical and extremal properties. PhD thesis, Technische Universität, Berlin (2007)Google Scholar
  12. 12.
    Goemans, M.: Smallest compact formulation for the permutahedron. http://www-math.mit.edu/goemans/publ.html (2009)
  13. 13.
    Kaibel, V., Pashkovich, K.: Constructing extended formulations from reflection relations. In: Günlük, O., Woeginger, G. (eds.) Integer Programming and Combinatorial Optimization. Proceedings of IPCO XV, New York, NY volume 6655 of Lecture Notes in Computer Science, pp. 287–300. Springer, Berlin (2011)Google Scholar
  14. 14.
    Kaibel, V., Pashkovich, K., Theis, D.O.: Symmetry matters for sizes of extended formulations. SIAM J. Discrete Math. 26(3), 1361–1382 (2012)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Kaibel, V., Walter, M.: Simple extensions of polytopes. In: Lee, J., Vygen, J. (eds.) Integer Programming and Combinatorial Optimization. Proceedings of IPCO XVII, Bonn, volume 8494 of Lecture Notes in Computer Science. Springer, Berlin (2014)Google Scholar
  16. 16.
    Kipp Martin, R.: Using separation algorithms to generate mixed integer model reformulations. Oper. Res. Lett. 10(3), 119–128 (1991)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Padberg, M.W., Rao, M.R.: The travelling salesman problem and a class of polyhedra of diameter two. Math. Program. 7, 32–45 (1974). doi: 10.1007/BF01585502 MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Pashkovich, K.: Tight lower bounds on the sizes of symmetric extensions of permutahedra and similar results. Math. Oper. Res. 39(4), 1330–1339 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Pokutta, S., Van Vyve, M.: A note on the extension complexity of the knapsack polytope. Oper. Res. Lett. 41(4), 347–350 (2013)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Rothvoss, T.: Some 0/1 polytopes need exponential size extended formulations. Math. Program., Ser. A 142, 255–268 (2013)Google Scholar
  21. 21.
    Rothvoss, T.: The matching polytope has exponential extension complexity. In: Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC’14, New York, NY, USA, pp. 263–272. ACM, New York (2014)Google Scholar
  22. 22.
    Santos, F.: A counterexample to the hirsch conjecture. Ann. Math. 176(1), 383–412 (2012)MATHCrossRefGoogle Scholar
  23. 23.
    Schrijver, A.: Combinatorial Optimization—Polyhedra and Efficiency. Springer, Berlin (2003)MATHGoogle Scholar
  24. 24.
    Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Ziegler, G.M.: Lectures on Polytopes (Graduate Texts in Mathematics). Springer, Berlin (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2015

Authors and Affiliations

  1. 1.Otto-von-Guericke Universität Magdeburg (FMA/IMO)MagdeburgGermany

Personalised recommendations