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Sum-of-Squares Rank Upper Bounds for Matching Problems

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Combinatorial Optimization (ISCO 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9849))

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Abstract

The matching problem is one of the most studied combinatorial optimization problems in the context of extended formulations and convex relaxations. In this paper we provide upper bounds for the rank of the Sum-of-Squares (SoS)/Lasserre hierarchy for a family of matching problems. In particular, we show that when the problem formulation is strengthened by incorporating the objective function in the constraints, the hierarchy requires at most \(\lceil \frac{k}{2} \rceil \) rounds to refute the existence of a perfect matching in an odd clique of size \(2k+1\).

Supported by the Swiss National Science Foundation project 200020-144491/1 “Approximation Algorithms for Machine Scheduling Through Theory and Experiments”.

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Notes

  1. 1.

    An r-uniform hypergraph is given by a set of vertices V and set of hyperedges E where each hyperedge \(e \in E\) is incident to exactly r vertices.

References

  1. Au, Y.H., Tunçel, L.: Complexity analyses of Bienstock–Zuckerberg and Lasserre relaxations on the matching and stable set polytopes. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 14–26. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  2. Au, Y., Tunçel, L.: A comprehensive analysis of polyhedral lift-and-project methods (2013). CoRR, abs/1312.5972

    Google Scholar 

  3. Avis, D., Bremner, D., Tiwary, H.R., Watanabe, O.: Polynomial size linear programs for non-bipartite matching problems and other problems in P (2014). CoRR, abs/1408.0807

    Google Scholar 

  4. Barak, B., Kelner, J.A., Steurer, D.: Rounding sum-of-squares relaxations. In: STOC, pp. 31–40 (2014)

    Google Scholar 

  5. Braun, G., Brown-Cohen, J., Huq, A., Pokutta, S., Raghavendra, P., Roy, A., Weitz, B., Zink, D.: The matching problem has no small symmetric SDP. In: SODA, pp. 1067–1078 (2016)

    Google Scholar 

  6. Chan, S.O., Lee, J.R., Raghavendra, P., Steurer, D.: Approximate constraint satisfaction requires large LP relaxations. In: FOCS, pp. 350–359. IEEE Computer Society (2013)

    Google Scholar 

  7. Edmonds, J.: Paths, trees and flowers. Can. J. Math. 17, 449–467 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fawzi, H., Saunderson, J., Parrilo, P.A.: Sparse sums of squares on finite abelian groups and improved semidefinite lifts. Math. Program., 1–43 (2016)

    Google Scholar 

  9. Goemans, M.X., Tunçel, L.: When does the positive semidefiniteness constraint help in lifting procedures? Math. Oper. Res. 26(4), 796–815 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grigoriev, D.: Linear lower bound on degrees of Positivstellensatz calculus proofs for the parity. Theoret. Comput. Sci. 259(1–2), 613–622 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Karlin, A.R., Mathieu, C., Nguyen, C.T.: Integrality gaps of linear and semi-definite programming relaxations for Knapsack. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 301–314. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  12. Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Laurent, M.: A comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre relaxations for 0–1 programming. Math. Oper. Res. 28(3), 470–496 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lee, J.R., Raghavendra, P., Steurer, D.: Lower bounds on the size of semidefinite programming relaxations. In: Servedio, R.A., Rubinfeld, R. (eds.) STOC, pp. 567–576. ACM (2015)

    Google Scholar 

  15. Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1(12), 166–190 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mathieu, C., Sinclair, A.: Sherali-Adams relaxations of the matching polytope. In: STOC, pp. 293–302 (2009)

    Google Scholar 

  17. Nesterov, Y.: Global Quadratic Optimization via Conic Relaxation, pp. 363–384. Kluwer Academic Publishers, Dordrecht (2000)

    Google Scholar 

  18. O’Donnell, R., Zhou, Y.: Approximability and proof complexity. In: SODA, pp. 1537–1556 (2013)

    Google Scholar 

  19. Parrilo, P.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. thesis, California Institute of Technology (2000)

    Google Scholar 

  20. Rothvoß, T.: The Lasserre Hierarchy in Approximation Algorithms – Lecture Notes for the MAPSP 2013 Tutorial, June 2013

    Google Scholar 

  21. Rothvoß, T.: The matching polytope has exponential extension complexity. In: STOC, pp. 263–272 (2014)

    Google Scholar 

  22. Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math. 3(3), 411–430 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Shor, N.: Class of global minimum bounds of polynomial functions. Cybernetics 23(6), 731–734 (1987)

    Article  MATH  Google Scholar 

  24. Stephen, T., Tunçel, L.: On a representation of the matching polytope via semidefinite liftings. Math. Oper. Res. 24(1), 1–7 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Worah, P.: Rank bounds for a hierarchy of Lovász and Schrijver. J. Comb. Optim. 30(3), 689–709 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yannakakis, M.: 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. IDSIA, 6928, Manno, Switzerland

    Adam Kurpisz, Samuli Leppänen & Monaldo Mastrolilli

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  1. Adam Kurpisz
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  2. Samuli Leppänen
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  3. Monaldo Mastrolilli
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Correspondence to Samuli Leppänen .

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Editors and Affiliations

  1. University of Salerno , Fisciano, Italy

    Raffaele Cerulli

  2. Kyoto University , Kyoto, Japan

    Satoru Fujishige

  3. LAMSADE, Université Paris-Dauphine , Paris , France

    A. Ridha Mahjoub

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Kurpisz, A., Leppänen, S., Mastrolilli, M. (2016). Sum-of-Squares Rank Upper Bounds for Matching Problems. In: Cerulli, R., Fujishige, S., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2016. Lecture Notes in Computer Science(), vol 9849. Springer, Cham. https://doi.org/10.1007/978-3-319-45587-7_35

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  • DOI: https://doi.org/10.1007/978-3-319-45587-7_35

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45586-0

  • Online ISBN: 978-3-319-45587-7

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