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Mathematical Programming

, Volume 153, Issue 1, pp 67–74 | Cite as

Tropical lower bounds for extended formulations

  • Yaroslav Shitov
Full Length Paper Series B

Abstract

The tropical arithmetic operations on \(\mathbb {R}\), defined as \(\oplus :(a,b)\rightarrow \min \{a,b\}\) and \(\otimes :(a,b)\rightarrow a+b\), arise from studying the geometry over non-Archimedean fields. We present an application of tropical methods to the study of extended formulations for convex polytopes. We propose a non-Archimedean generalization of the well known Boolean rank bound for the extension complexity. We show how to construct a real polytope with the same extension complexity and combinatorial type as a given non-Archimedean polytope. Our results allow us to develop a method of constructing real polytopes with large extension complexity.

Keywords

Convex polytope Extended formulation Tropical mathematics 

Mathematics Subject Classification

15A23 15B48 52B05 

References

  1. 1.
    Arora, S., Ge, R., Kannan, R., Moitra, A.: Computing a nonnegative matrix factorization—provably. In: Proceedings of the 44th Symposium on Theory of Computing, pp. 145–162, ACM (2012)Google Scholar
  2. 2.
    Barvinok, A.I.: Two algorithmic results for the traveling salesman problem. Math. Oper. Res. 21(1), 65–84 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Cartwright, D., Chan, M.: Three notions of tropical rank for symmetric matrices. Combinatorica 32(1), 55–84 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Cohen, J.E., Rothblum, U.G.: Nonnegative ranks, decompositions, and factorizations of nonnegative matrices. Linear Algebra Appl. 190, 149–168 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Conforti, M., Cornuejols, G., Zambelli, G.: Extended formulations in combinatorial optimization. 4OR 8(1), 1–48 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Develin, M.: Tropical secant varieties of linear spaces. Discreat. Comp. Geom. 35(1), 117–129 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Develin, M., Santos, F., Sturmfels, B.: On the rank of a tropical matrix. In: Goodman, E., Pach, J., Welzl, E. (eds.) Discrete and Computational Geometry. MSRI Publications, Cambridge University Press, Cambridge (2005)Google Scholar
  8. 8.
    Engeler, E.: Metamathematik der Elementarmathematik. Springer, Berlin (1983)zbMATHCrossRefGoogle Scholar
  9. 9.
    Fiorini, S., Rothvoß, T., Tiwary, H. R.: Extended formulations for polygons. Discreat. Comp. Geom. 48(3), 1–11 (2012)Google Scholar
  10. 10.
    Guterman, A., Shitov, Ya.: Tropical patterns of matrices and the Gondran–Minoux rank function. Linear Algebra Appl. 437, 1793–1811 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Gouveia, J., Parillo, P. A., Thomas, R.R.: Lifts of convex sets and cone factorizations. Math. Oper. Res. 38, 248–264 (2013)Google Scholar
  12. 12.
    Gouveia, J., Robinson, R. Z., Thomas, R. R.: Polytopes of Minimum Positive Semidefinite Rank. Arxiv preprint arXiv:1205.5306
  13. 13.
    Gillis, N., Glineur, F.: On the geometric interpretation of the nonnegative rank. Linear Algebra Appl. 437, 2685–2712 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Kaibel, V.: Extended formulations in combinatorial optimization. Optima 85, 2–7 (2011)Google Scholar
  15. 15.
    Rayner, F. J.: Algebraically closed fields analogous to fields of Puiseux series. J. Lond. Math. Soc. 8, 504–506 (1974)Google Scholar
  16. 16.
    Shitov, Y.: The complexity of tropical matrix factorization. Adv. Math. 254, 138–156 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Shitov, Y.: An upper bound for nonnegative rank. J. Comb. Theory Ser. A 122, 126–132 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Shitov, Y.: Studying nonnegative factorizations with tools from linear algebra over a semiring. Comm. Algebra (to appear)Google Scholar
  19. 19.
    Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. Comput. Syst. Sci. 43, 441–466 (1991)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia

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