Discrete & Computational Geometry

, Volume 53, Issue 2, pp 397–401 | Cite as

A Short Proof that the Extension Complexity of the Correlation Polytope Grows Exponentially

  • Volker Kaibel
  • Stefan Weltge


We establish that the extension complexity of the \(n\times n\) correlation polytope is at least \(1.5\,^n\) by a short proof that is self-contained except for using the fact that every face of a polyhedron is the intersection of all facets it is contained in. The main innovative aspect of the proof is a simple combinatorial argument showing that the rectangle covering number of the unique-disjointness matrix is at least \(1.5^n\), and thus the nondeterministic communication complexity of the unique-disjointness predicate is at least \(.58n\). We thereby slightly improve on the previously best known lower bounds \(1.24^n\) and \(.31n\), respectively.


Correlation polytope Extended formulations Unique disjointness Communication complexity 

Mathematics Subject Classification

52Bxx 90C57 94Axx 



We thank Yuri Faenza and Kanstantsin Pashkovich for their helpful remarks on an earlier version of this paper. Part of this research was funded by the German Research Foundation (DFG): “Extended Formulations in Combinatorial Optimization” (KA 1616/4-1).


  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. Lecture Notes in Computer Science, vol. 7965, pp. 57–68. Springer, Berlin (2013)CrossRefGoogle Scholar
  2. 2.
    Braun, G., Pokutta, S.: Common information and unique disjointness. In: Foundations of Computer Science (FOCS), IEEE 54th Annual Symposium, pp. 688–697 (2013)Google Scholar
  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.
    Jukna, S.: Boolean Function Complexity: Advances and Frontiers. Springer, Berlin (2012)CrossRefGoogle Scholar
  5. 5.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (2006)Google Scholar
  6. 6.
    Pokutta, S., Vyve, M.V.: A note on the extension complexity of the knapsack polytope. Oper. Res. Lett. 41, 347–350 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Razborov, A.A.: On the distributional complexity of disjointness. Theor. Comput. Sci. 106(2), 385–390 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Rothvoß, T.: The matching polytope has exponential extension complexity. (2013)
  9. 9.
    de Wolf, R.: Nondeterministic quantum query and communication complexities. SIAM J. Comput. 32(3), 681–699 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43(3), 441–466 (1991)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

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

Personalised recommendations