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Extended Formulations, Nonnegative Factorizations, and Randomized Communication Protocols

  • Yuri Faenza
  • Samuel Fiorini
  • Roland Grappe
  • Hans Raj Tiwary
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7422)

Abstract

We show that the binary logarithm of the nonnegative rank of a nonnegative matrix is, up to small constants, equal to the minimum complexity of a randomized communication protocol computing the matrix in expectation. We use this connection to prove new conditional lower bounds on the sizes of extended formulations, in particular, for perfect matching polytopes.

Keywords

Extended formulations Nonnegative rank Communication protocols 

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References

  1. 1.
    Babai, L., Frankl, P., Simon, J.: Complexity classes in communication complexity theory. In: 27th Annual Symposium on Foundations of Computer Science (FOCS 1986), pp. 337–347. IEEE Computer Society Press, Toronto (1986)CrossRefGoogle Scholar
  2. 2.
    Chvátal, V.: On certain polytopes associated with graphs. J. Comb. Theory B 18, 138–154 (1975)zbMATHCrossRefGoogle Scholar
  3. 3.
    Cohen, J.E., Rothblum, U.G.: Nonnegative ranks, decompositions, and factorizations of nonnegative matrices. Linear Algebra Appl. 190, 149–168 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Conforti, M., Cornuéjols, G., Zambelli, G.: Extended formulations in combinatorial optimization. 4OR 8(1), 1–48 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Edmonds, J.: Maximum matching and a polyhedron with 0, 1 vertices. J. Res. Nat. Bur. Stand 69B, 125–130 (1965)MathSciNetGoogle Scholar
  6. 6.
    Edmonds, J.: Matroids and the greedy algorithm. Math. Program. 1, 127–136 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Fiorini, S., Kaibel, V., Pashkovich, K., Theis, D.O.: Combinatorial bounds on nonnegative rank and extended formulations, arXiv:1111.0444Google Scholar
  8. 8.
    Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., de Wolf, R.: Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds. In: Proceedings of the 44th ACM Symposium on Theory of Computing, STOC 2012 (to appear, 2012)Google Scholar
  9. 9.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric algorithms and combinatorial optimization. In: Algorithms and Combinatorics, 2nd edn., vol. 2. Springer, Berlin (1993)Google Scholar
  10. 10.
    Kaibel, V.: Extended formulations in combinatorial optimization. Optima 85, 2–7 (2011)Google Scholar
  11. 11.
    Kaibel, V., Pashkovich, K., Theis, D.O.: Symmetry Matters for the Sizes of Extended Formulations. In: Eisenbrand, F., Shepherd, F.B. (eds.) IPCO 2010. LNCS, vol. 6080, pp. 135–148. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Kalyanasundaram, B., Schnitger, G.: The probabilistic communication complexity of set intersection. SIAM J. Discr. Math. 5(4), 545–557 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Kushilevitz, E., Nisan, N.: Communication complexity. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  14. 14.
    Martin, R.K.: Using separation algorithms to generate mixed integer model reformulations. Oper. Res. Lett. 10(3), 119–128 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Razborov, A.A.: On the distributional complexity of disjointness. Theor. Comput. Sci. 106(2), 385–390 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Schrijver, A.: Combinatorial optimization. Polyhedra and efficiency. Algorithms and Combinatorics, vol. 24(A, B). Springer, Berlin (2003)Google Scholar
  17. 17.
    Vazirani, V.V.: Approximation algorithms. Springer (2001)Google Scholar
  18. 18.
    Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. System Sci. 43(3), 441–466 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, Berlin (1995)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yuri Faenza
    • 1
  • Samuel Fiorini
    • 2
  • Roland Grappe
    • 3
  • Hans Raj Tiwary
    • 2
  1. 1.Dipartimento di Matematica Pura e ApplicataUniversità di PadovaPadovaItaly
  2. 2.Département de MathématiqueUniversité Libre de BruxellesBrusselsBelgium
  3. 3.Laboratoire d’Informatique de Paris-Nord, UMR CNRS 7030, Institut GaliléeUniversité Paris-NordVilletaneuseFrance

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