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Complexity of combinatorial optimization problems in terms of face lattices of associated polytopes

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Abstract

This paper deals with the following question: Can combinatorial properties of polytopes help in finding an estimate for the complexity of the corresponding optimization problem? Sometimes, these key characteristics of complexity were the number of hyperfaces of the polytope, diameter and clique number of the graph of the polytope, the rectangle covering number of the vertex-facet incidence matrix, and some other characteristics. In this paper, we provide several families of polytopes for which the above-mentioned characteristics differ significantly from the real computational complexity of the corresponding optimization problems. In particular, we give two examples of discrete optimization problem whose polytopes are combinatorially equivalent and they have the same lengths of the binary representation of the coordinates of the polytope vertices. Nevertheless, the first problem is solvable in polynomial time, while the second problem has exponential complexity.

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References

  1. V. A. Bondarenko and A. N. Maksimenko, Geometric Constructions and Complexity in Combinatorial Optimization (LKI, Moscow, 2008) [in Russian].

    Google Scholar 

  2. V. A. Bondarenko and A. V. Nikolaev, “Combinatorial and Geometric Properties of the Max-Cut and Min-Cut Problems,” Dokl. Akad. Nauk 452 (2), 127–129 (2013) [Dokl. Math. 88 (2), 516–517 (2013)].

    MathSciNet  MATH  Google Scholar 

  3. M. M. Deza and M. Laurent, Geometry of Cuts and Metrics (Springer, Heidelberg, 1997; MTsNMO, Moscow, 2001).

    MATH  Google Scholar 

  4. A. N. Maksimenko, “The Common Face of Some 0/1-Polytopes with NP-Complete Nonadjacency Relation,” Fundam. Prikl. Mat. 18 (2), 105–118 (2013) [J. Math. Sci. 203 (6), 823–832 (2014)].

    Google Scholar 

  5. A. N. Maksimenko, “Characteristics of Complexity: Clique Number of a Polytope Graph and Rectangle Covering Number,” Model. Anal. Inform. Sist. 21 (5), 116–130 (2014).

    Google Scholar 

  6. A. N. Maksimenko, “The Simplest Families of Polytopes Associated with NP-Hard Problems,” Dokl. Akad. Nauk 460 (3), 272–274 (2015) [Dokl. Math. 91 (1), 53–55 (2015)].

    MathSciNet  MATH  Google Scholar 

  7. G. M. Ziegler, Lectures on Polytopes (Springer, New York, 1995; MTsNMO, Moscow, 2014).

    MATH  Google Scholar 

  8. D. L. Applegate, R. M. Bixby, V. Chvátal, and W. J. Cook, The Traveling Salesman Problem: A Computational Study (Princeton Univ. Press, Princeton, 2007).

    MATH  Google Scholar 

  9. Yu. Bogomolov, S. Fiorini, A. N. Maksimenko, and K. Pashkovich, “Small Extended Formulations for Cyclic Polytopes,” Discrete Comput. Geom. 53 (4), 809–816 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Conforti, G. Cornuéjols, and G. Zambelli, “Extended Formulations in CombinatorialOptimization,” Ann. Oper. Res. 204 (1), 97–143 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  11. G. B. Dantzig, D. R. Fulkerson, and S. M. Johnson, “Solution of a Large-Scale Traveling Salesman Problem,” Oper. Res. 2 (4), 393–410 (1954).

    MathSciNet  Google Scholar 

  12. S. Fiorini, V. Kaibel, K. Pashkovich, and D. O. Theis, “Combinatorial Bounds on Nonnegative Rank and Extended Formulations,” DiscreteMath. 313 (1), 67–83 (2013).

    MathSciNet  MATH  Google Scholar 

  13. S. Fiorini, S. Massar, S. Pokutta, H. R. Tiwary, and R. deWolf, “Exponential Lower Bounds for Polytopes in Combinatorial Optimization,” J. ACM 62 (2), 17:1–17:23 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Fiorini, T. Rothvoß, and H. R. Tiwary, “Extended Formulations for Polygons,” Discrete Comput. Geom. 48 (13), 658–668 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  15. B. Grünbaum, Convex Polytopes (Springer, New York, 2003).

    Book  MATH  Google Scholar 

  16. V. Kaibel, “Extended Formulations in Combinatorial Optimization,” Optima 85, Math. Optim. Soc. Newsl. No. 85, 2–7 (2011).

    Google Scholar 

  17. V. Kaibel and M. E. Pfetsch, “Computing the Face Lattice of a Polytope from Its Vertex-Facet Incidences,” Comput. Geom. 23 (3), 281–290 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  18. V. Kaibel and S. Weltge, “A Short Proof That the Extension Complexity of the Correlation Polytope Grows Exponentially,” Discrete Comput. Geom. 53 (2), 397–401 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  19. M. W. Padberg and M. R. Rao, “The Travelling Salesman Problem and a Class of Polyhedra of Diameter Two,” Math. Program. 7, 32–45 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  20. T. Rothvoß, “The Matching Polytope Has Exponential Extension Complexity,” in Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC 2014), New York, May 31–June 3, 2014 (ACM, New York, 2014), pp. 263–272.

    Google Scholar 

  21. C. Shannon, “Communication Theory of Secrecy Systems,” Bell System Techn. J. 28 (4), 656–715 (1949).

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Yannakakis, “Expressing Combinatorial Optimization Problems by Linear Programs,” J. Comput. Syst. Sci. 43 (3), 441–466 (1991).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Demidov Yaroslavl’ State University, ul. Sovetskaya 14, Yaroslavl’, 150000, Russia

    A. N. Maksimenko

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  1. A. N. Maksimenko
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Correspondence to A. N. Maksimenko.

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Original Russian Text © A.N. Maksimenko, 2016, published in Diskretnyi Analiz i Issledovanie Operatsii, 2016, Vol. 23, No. 3, pp. 61–80.

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Maksimenko, A.N. Complexity of combinatorial optimization problems in terms of face lattices of associated polytopes. J. Appl. Ind. Math. 10, 370–379 (2016). https://doi.org/10.1134/S1990478916030078

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  • DOI: https://doi.org/10.1134/S1990478916030078

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