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.
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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)
Barvinok, A.I.: Two algorithmic results for the traveling salesman problem. Math. Oper. Res. 21(1), 65–84 (1996)
Cartwright, D., Chan, M.: Three notions of tropical rank for symmetric matrices. Combinatorica 32(1), 55–84 (2012)
Cohen, J.E., Rothblum, U.G.: Nonnegative ranks, decompositions, and factorizations of nonnegative matrices. Linear Algebra Appl. 190, 149–168 (1993)
Conforti, M., Cornuejols, G., Zambelli, G.: Extended formulations in combinatorial optimization. 4OR 8(1), 1–48 (2010)
Develin, M.: Tropical secant varieties of linear spaces. Discreat. Comp. Geom. 35(1), 117–129 (2006)
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)
Engeler, E.: Metamathematik der Elementarmathematik. Springer, Berlin (1983)
Fiorini, S., Rothvoß, T., Tiwary, H. R.: Extended formulations for polygons. Discreat. Comp. Geom. 48(3), 1–11 (2012)
Guterman, A., Shitov, Ya.: Tropical patterns of matrices and the Gondran–Minoux rank function. Linear Algebra Appl. 437, 1793–1811 (2012)
Gouveia, J., Parillo, P. A., Thomas, R.R.: Lifts of convex sets and cone factorizations. Math. Oper. Res. 38, 248–264 (2013)
Gouveia, J., Robinson, R. Z., Thomas, R. R.: Polytopes of Minimum Positive Semidefinite Rank. Arxiv preprint arXiv:1205.5306
Gillis, N., Glineur, F.: On the geometric interpretation of the nonnegative rank. Linear Algebra Appl. 437, 2685–2712 (2012)
Kaibel, V.: Extended formulations in combinatorial optimization. Optima 85, 2–7 (2011)
Rayner, F. J.: Algebraically closed fields analogous to fields of Puiseux series. J. Lond. Math. Soc. 8, 504–506 (1974)
Shitov, Y.: The complexity of tropical matrix factorization. Adv. Math. 254, 138–156 (2014)
Shitov, Y.: An upper bound for nonnegative rank. J. Comb. Theory Ser. A 122, 126–132 (2014)
Shitov, Y.: Studying nonnegative factorizations with tools from linear algebra over a semiring. Comm. Algebra (to appear)
Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. Comput. Syst. Sci. 43, 441–466 (1991)
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Shitov, Y. Tropical lower bounds for extended formulations. Math. Program. 153, 67–74 (2015). https://doi.org/10.1007/s10107-014-0833-6
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DOI: https://doi.org/10.1007/s10107-014-0833-6