Semidefinite Programming and Approximation Algorithms: A Survey

  • Sanjeev Arora
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7074)

Abstract

Computing approximate solutions for NP-hard problems is an important research endeavor. Since the work of Goemans-Williamson in 1993, semidefinite programming (a form of convex programming in which the variables are vector inner products) has been used to design the current best approximation algorithms for problems such as MAX-CUT, MAX-3SAT, SPARSEST CUT, GRAPH COLORING, etc. The talk will survey this area, as well as its fascinating connections with topics such as geometric embeddings of metric spaces, and Khot’s unique games conjecture.

The talk will be self-contained.

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References

  1. 1.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press (2009)Google Scholar
  2. 2.
    Arora, S., Lee, J.R., Naor, A.: Euclidean distortion and the sparsest cut. J. Amer. Math. Soc. 21(1), 1–21 (2008)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Arora, S., Lund, C.: Hardness of approximations. In: Approximation Algorithms for NP-hard Problems. Course Technology (1996)Google Scholar
  4. 4.
    Arora, S., Rao, S., Vazirani, U.V.: Expander flows, geometric embeddings and graph partitioning. J. ACM 56(2) (2009); Prelim version ACM STOC 2004Google Scholar
  5. 5.
    Barak, B., Raghavendra, P., Steurer, D.: Rounding semidefinite programming hierarchies via global correlation. CoRR, abs/1104.4680 (2011); To appear in IEEE FOCS 2011Google Scholar
  6. 6.
    Chawla, S., Gupta, A., Räcke, H.: Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut. In: SODA 2005: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 102–111. Society for Industrial and Applied Mathematics, Philadelphia (2005)Google Scholar
  7. 7.
    Goemans, M.X.: Semidefinite programming and combinatorial optimization. In: Proceedings of the International Congress of Mathematicians, Berlin, vol. III, pp. 657–666 (1998)Google Scholar
  8. 8.
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. Assoc. Comput. Mach. 42(6), 1115–1145 (1995)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Harsha, P., Charikar, M., Andrews, M., Arora, S., Khot, S., Moshkovitz, D., Zhang, L., Aazami, A., Desai, D., Gorodezky, I., Jagannathan, G., Kulikov, A.S., Mir, D.J., Newman, A., Nikolov, A., Pritchard, D., Spencer, G.: Limits of approximation algorithms: Pcps and unique games (dimacs tutorial lecture notes). CoRR, abs/1002.3864 (2010)Google Scholar
  10. 10.
    Khot, S.: On the unique games conjecture (invited survey). In: Annual IEEE Conference on Computational Complexity, pp. 99–121 (2010)Google Scholar
  11. 11.
    Raghavendra, P., Steurer, D.: Integrality gaps for strong SDP relaxations of Unique Games. In: FOCS, pp. 575–585 (2009)Google Scholar
  12. 12.
    Shmoys, D., Williamson, D.: Design of Approximation Algorithms. Cambridge University Press (2011)Google Scholar
  13. 13.
    Trevisan, L.: Inapproximability of combinatorial optimization problems. CoRR, cs.CC/0409043 (2004)Google Scholar
  14. 14.
    Trevisan, L.: Inapproximability of combinatorial optimization problems. CoRR, cs.CC/0409043 (2004)Google Scholar
  15. 15.
    Vazirani, V.: Approximation Algorithms. Springer, Heidelberg (2001)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Sanjeev Arora
    • 1
  1. 1.Computer SciencePrinceton UniversityPrincetonUSA

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