Extension Complexity and Realization Spaces of Hypersimplices

Article

Abstract

The (nk)-hypersimplex is the convex hull of all 0 / 1-vectors of length n with coordinate sum k. We explicitly determine the extension complexity of all hypersimplices as well as of certain classes of combinatorial hypersimplices. To that end, we investigate the projective realization spaces of hypersimplices and their (refined) rectangle covering numbers. Our proofs combine ideas from geometry and combinatorics and are partly computer assisted.

Keywords

Hypersimplices Extension complexity Nonnegative rank Rectangle covering number Realization spaces 

Mathematics Subject Classification

90C57 52B12 15A23 

Supplementary material

454_2017_9925_MOESM1_ESM.py (5 kb)
Supplementary material 1 (py 4 KB)
454_2017_9925_MOESM2_ESM.py (4 kb)
Supplementary material 2 (py 3 KB)

References

  1. 1.
    Barnette, D., Grünbaum, B.: Preassigning the shape of a face. Pac. J. Math. 32, 299–306 (1970)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Beasley, L.B., Klauck, H., Lee, T., Theis, D.O.: Communication complexity, linear optimization, and lower bounds for the nonnegative rank of matrices (Dagstuhl Seminar 13082). Dagstuhl Rep. 3(2), 127–143 (2013)Google Scholar
  3. 3.
    Belov, A., Heule, M.J.H., Järvisalo, M.: Proceedings of SAT Competition 2014. Department of Computer Science Series of Publications B, vol. B-2014-2. University of Helsinki, Helsinki (2014). http://fmv.jku.at/lingeling
  4. 4.
    Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.M.: Oriented Matroids. Encyclopedia of Mathematics and Its Applications, vol. 46, 2nd edn. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  5. 5.
    Fiorini, S., Kaibel, V., Pashkovich, K., Theis, D.O.: Combinatorial bounds on nonnegative rank and extended formulations. Discrete Math. 313(1), 67–83 (2013)Google Scholar
  6. 6.
    Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., de Wolf, R.: Exponential lower bounds for polytopes in combinatorial optimization. J. ACM 62(2), 17 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Fiorini, S., Rothvoß, T., Tiwary, H.R.: Extended formulations for polygons. Discrete Comput. Geom. 48(3), 658–668 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gawrilow, E., Joswig, M.: Polymake: a framework for analyzing convex polytopes. In: Kalai, G., Ziegler, G.M. (eds.) Polytopes—Combinatorics and Computation. DMV Seminar, vol. 29, pp. 43–74. Birkhäuser, Basel (2000)Google Scholar
  9. 9.
    Gel’fand, I.M., Goresky, R.M., MacPherson, R.D., Serganova, V.V.: Combinatorial geometries, convex polyhedra, and Schubert cells. Adv. Math. 63(3), 301–316 (1987)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gouveia, J., Robinson, R.Z., Thomas, R.R.: Polytopes of minimum positive semidefinite rank. Discrete Comput. Geom. 50(3), 679–699 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Grünbaum, B.: Convex Polytopes. Graduate Texts in Mathematics, vol. 221, 2nd edn. Springer, New York (2003)CrossRefMATHGoogle Scholar
  12. 12.
    Grande, F., Sanyal, R.: Theta rank, levelness, and matroid minors. J. Comb. Theory Ser. B 123, 1–31 (2017)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Klauck, H., Lee, T., Theis, D.O., Thomas, R.R.: Limitations of convex programming: lower bounds on extended formulations and factorization ranks (Dagstuhl Seminar 15082). Dagstuhl Rep. 5(2), 109–127 (2015)Google Scholar
  14. 14.
    Kaibel, V., Weltge, S.: A short proof that the extension complexity of the correlation polytope grows exponentially. Discrete Comput. Geom. 53(2), 396–401 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Mnëv, N.E.: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In: Viro, O.Y., Vershik, A.M. (eds.) Topology and Geometry—Rohlin Seminar. Lecture Notes in Mathematics, vol. 1346, pp. 527–543. Springer, Berlin (1988)CrossRefGoogle Scholar
  16. 16.
    Oelze, M., Vandaele, A., Weltge, S.: Computing the extension complexities of all 4-dimensional 0/1-polytopes. http://arxiv.org/abs/1406.4895 (2014)
  17. 17.
    Oxley, J.G.: Matroid Theory. Oxford Graduate Texts in Mathematics, vol. 21, 2nd edn. Oxford University Press, Oxford (2011)CrossRefMATHGoogle Scholar
  18. 18.
    Padrol, A.: Extension complexity of polytopes with few vertices or facets. SIAM J. Discrete Math. 30(4), 2162–2176 (2016)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Padrol, A., Pfeifle, J.: Polygons as sections of higher-dimensional polytopes. Electron. J. Comb. 22(1), 1.24-1–1.24-16 (2015)MathSciNetMATHGoogle Scholar
  20. 20.
    Richter-Gebert, J.: Realization Spaces of Polytopes. Lecture Notes in Mathematics, vol. 1643. Springer, Berlin (1996)MATHGoogle Scholar
  21. 21.
    Rothvoß, T.: Some 0/1 polytopes need exponential size extended formulations. Math. Program. 142(1–2), 255–268 (2013)Google Scholar
  22. 22.
    Rothvoß, T.: The matching polytope has exponential extension complexity. In: Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC’14), pp. 263–272. ACM, New York (2014)Google Scholar
  23. 23.
    Shitov, Ya.: Sublinear extensions of polygons. http://arxiv.org/abs/1412.0728 (2014)
  24. 24.
    Shitov, Ya.: An upper bound for nonnegative rank. J. Comb. Theory Ser. A 122, 126–132 (2014)Google Scholar
  25. 25.
    Wheeler, A.K.: Ideals generated by principal minors. http://arxiv.org/abs/1410.1910 (2015)
  26. 26.
    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathemtaics, vol. 152. Springer, New York (1995)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Fachbereich Mathematik und InformatikFreie Universität BerlinBerlinGermany
  2. 2.Sorbonne Universités, Université Pierre et Marie Curie (Paris 6), Institut de Mathématiques de Jussieu – Paris Rive Gauche (UMR 7586)ParisFrance
  3. 3.Institut für MathematikGoethe-Universität FrankfurtFrankfurt am MainGermany

Personalised recommendations