On Vertices and Facets of Combinatorial 2-Level Polytopes

  • Manuel AprileEmail author
  • Alfonso Cevallos
  • Yuri Faenza
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9849)


2-level polytopes naturally appear in several areas of mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics.

We investigate upper bounds on the product of the number of facets \(f_{d-1}(P)\) and the number of vertices \(f_0(P)\), where d is the dimension of a 2-level polytope P. This question was first posed in [3], where experimental results showed \(f_0(P)f_{d-1}(P)\le d 2^{d+1}\) up to \(d=6\).

We show that this bound holds for all known (to the best of our knowledge) 2-level polytopes coming from combinatorial settings, including stable set polytopes of perfect graphs and all 2-level base polytopes of matroids. For the latter family, we also give a simple description of the facet-defining inequalities. These results are achieved by an investigation of related combinatorial objects, that could be of independent interest.


Perfect Graph Linear Description Cycle Space Binary Matroids Eulerian Subgraph 
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  1. 1.
    Barahona, F., Grötschel, M.: On the cycle polytope of a binary matroid. J. Comb. Theory, Ser. B 40(1), 40–62 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bárány, I., Pór, A.: On 0-1 polytopes with many facets. Adv. Math. 161(2), 209–228 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bohn, A., Faenza, Y., Fiorini, S., Fisikopoulos, V., Macchia, M., Pashkovich, K.: Enumeration of 2-level polytopes. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 191–202. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  4. 4.
    Chaourar, B.: On the kth best base of a matroid. Oper. Res. Lett. 36(2), 239–242 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chvátal, V.: On certain polytopes associated with graphs. J. Comb. Theory Ser. B 18, 138–154 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 101. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  7. 7.
    Feichtner, E.M., Sturmfels, B.: Matroid polytopes, nested sets and bergman fans. Portugaliae Mathematica 62(4), 437–468 (2005)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Gouveia, J., Laurent, M., Parrilo, P.A., Thomas, R.: A new semidefinite programming hierarchy for cycles in binary matroids and cuts in graphs. Math. Program. 133(1–2), 203–225 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gouveia, J., Parrilo, P., Thomas, R.: Theta bodies for polynomial ideals. SIAM J. Optim. 20(4), 2097–2118 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grande, F., Rué, J.: Many 2-level polytopes from matroids. Discret. Comput. Geom. 54(4), 954–979 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grande, F., Sanyal, R.: Theta rank, levelness, and matroid minors (2014). arXiv:1408.1262
  12. 12.
    Gross, J.L., Yellen, J.: Graph Theory and Its Applications. CRC Press, Boca Raton (2005)zbMATHGoogle Scholar
  13. 13.
    Oxley, J.G.: Matroid Theory, vol. 3. Oxford University Press, Oxford (2006)zbMATHGoogle Scholar
  14. 14.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1998)zbMATHGoogle Scholar
  15. 15.
    Stanley, R.: Decompositions of rational convex polytopes. Ann. Discret. Math. 6, 333–342 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sullivant, S.: Compressed polytopes and statistical disclosure limitation. Tohoku Math. J. Second Ser. 58(3), 433–445 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43, 441–466 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ziegler, G.: Lectures on Polytopes, vol. 152. Springer, Berlin (1995)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.École Polytechnique Fédérale de LausanneLausanneSwitzerland

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