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Average case polyhedral complexity of the maximum stable set problem

Abstract

We study the minimum number of constraints needed to formulate random instances of the maximum stable set problem via linear programs (LPs), in two distinct models. In the uniform model, the constraints of the LP are not allowed to depend on the input graph, which should be encoded solely in the objective function. There we prove a \(2^{\Omega (n{/}\log n)}\) lower bound with probability at least \(1 - 2^{-2^n}\) for every LP that is exact for a randomly selected set of instances; each graph on at most n vertices being selected independently with probability \(p \geqslant 2^{- \left( {\begin{array}{c}n/4\\ 2\end{array}}\right) + n}\). In the non-uniform model, the constraints of the LP may depend on the input graph, but we allow weights on the vertices. The input graph is sampled according to the G(np) model. There we obtain upper and lower bounds holding with high probability for various ranges of p. We obtain a super-polynomial lower bound all the way from \(p = \Omega \left( \frac{\log ^{6+\varepsilon } n}{n} \right) \) to \(p = o\left( \frac{1}{\log n} \right) \). Our upper bound is close to this as there is only an essentially quadratic gap in the exponent, which currently also exists in the worst-case model. Finally, we state a conjecture that would close this gap, both in the average-case and worst-case models.

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References

  1. 1.

    Avis, D., Tiwary, H.R.: On the extension complexity of combinatorial polytopes. Math. Progr. B 153, 95–115 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Bazzi, A., Fiorini, S., Pokutta, S., Svensson, O.: Small linear programs cannot approximate Vertex Cover within a factor of \(2 - \varepsilon \). In: IEEE 56th Annual symposium on foundations of computer science (FOCS), pp. 1123–1142 (2015)

  3. 3.

    Braun, G., Pokutta, S.: Common information and unique disjointness. In: IEEE 54th Annual symposium on foundations of computer science (FOCS), pp. 688–697 (2013)

  4. 4.

    Braun, G., Pokutta, S.: The matching polytope does not admit fully-polynomial size relaxation schemes. In: Proceedings of the 26th annual ACM-SIAM symposium on discrete algorithms (SODA), pp. 837–846 (2015)

  5. 5.

    Braun, G., Fiorini, S., Pokutta, S., Steurer, D.: Approximation limits of linear programs (beyond hierarchies). Math. Oper. Res. 40, 756–772 (2015a)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Braun, G., Pokutta, S., Zink, D.: Inapproximability of combinatorial problems via small LPs and SDPs. In: Proceedings of the 47th annual ACM symposium on theory of computing (STOC), pp. 107–116 (2015b)

  7. 7.

    Braverman, M., Moitra, A.: An information complexity approach to extended formulations. In: Proceedings of the 45th annual ACM symposium on theory of computing (STOC), pp. 161–170 (2013)

  8. 8.

    Chan, S., Lee, J., Raghavendra, P., Steurer, D.: Approximate constraint satisfaction requires large LP relaxations. In: IEEE 54th annual symposium on foundations of computer science (FOCS 2013), pp. 350–359 (2013)

  9. 9.

    Conforti, M., Cornuéjols, G., Zambelli, G.: Extended formulations in combinatorial optimization. 4OR 8, 1–48 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Diestel, R.: Graph Theory. Springer, Heidelberg (2005)

    MATH  Google Scholar 

  11. 11.

    Edmonds, J.: Maximum matching and a polyhedron with 0, 1 vertices. J. Res. Natl. Bur. Stand. 69B, 125–130 (1965)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    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 annual ACM symposium on theory of computing (STOC) (2012)

  13. 13.

    Fiorini, S., Massar, S., Pokutta, S., Tiwary, H., Wolf, Rd: Exponential lower bounds for polytopes in combinatorial optimization. J. ACM 62(2), 17 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Kaibel, V.: Extended formulations in combinatorial optimization. Optima 85, 2–7 (2011)

    Google Scholar 

  16. 16.

    Kaibel, V., Weltge, S.: A short proof that the extension complexity of the correlation polytope grows exponentially. Discrete Comput. Geom. 53, 397–401 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Lee, J., Raghavendra, P., Steurer, D.: Lower bounds on the size of semidefinite programming relaxations. In: Proceedings of the 47th annual ACM symposium on theory of computing (STOC), pp. 567–576 (2015)

  18. 18.

    Pashkovich, K.: Extended Formulations for Combinatorial Polytopes. PhD thesis, Magdeburg Universität (2012)

  19. 19.

    Pokutta, S., Van Vyve, M.: A note on the extension complexity of the knapsack polytope. Oper. Res. Lett. 41, 347–350 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Razborov, A.A.: On the distributional complexity of disjointness. Theor. Comput. Sci. 106, 385–390 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Rothvoß, T.: Some 0/1 polytopes need exponential size extended formulations. Math. Progr. A 142, 255–268 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Rothvoß, T.: The matching polytope has exponential extension complexity. In: Proceedings of the 46th annual ACM symposium on theory of computing (STOC), pp. 263–272 (2014)

  23. 23.

    Williams, J.: A linear-size zero-one programming model for the minimum spanning tree problem in planar graphs. Networks 39, 53–60 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Wolf, Rd: Nondeterministic quantum query and communication complexities. SIAM J. Comput. 32(3), 681–699 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Yannakakis, M.: Expressing combinatorial optimization problems by linear programs (extended abstract). In: Proceedings of the 20th annual ACM symposium on theory of computing (STOC), pp. 223–228 (1988)

  26. 26.

    Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43, 441–466 (1991)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgments

Research reported in this paper was partially supported by NSF Grant CMMI-1300144.

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Correspondence to Gábor Braun.

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Braun, G., Fiorini, S. & Pokutta, S. Average case polyhedral complexity of the maximum stable set problem. Math. Program. 160, 407–431 (2016). https://doi.org/10.1007/s10107-016-0989-3

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Mathematics Subject Classification

  • Primary 68Q17
  • Secondary 05C69
  • 05C80
  • 90C05