A New Regularity Lemma and Faster Approximation Algorithms for Low Threshold Rank Graphs

  • Shayan Oveis Gharan
  • Luca Trevisan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8096)

Abstract

Kolla and Tulsiani [KT07, Kol11] and Arora, Barak and Steurer [ABS10] introduced the technique of subspace enumeration, which gives approximation algorithms for graph problems such as unique games and small set expansion; the running time of such algorithms is exponential in the threshold-rank of the graph.

Guruswami and Sinop [GS11, GS12], and Barak, Raghavendra, and Steurer [BRS11] developed an alternative approach to the design of approximation algorithms for graphs of bounded threshold-rank, based on semidefinite programming relaxations in the Lassere hierarchy and on novel rounding techniques. These algorithms are faster than the ones based on subspace enumeration and work on a broad class of problems.

In this paper we develop a third approach to the design of such algorithms. We show, constructively, that graphs of bounded threshold-rank satisfy a weak Szemeredi regularity lemma analogous to the one proved by Frieze and Kannan [FK99] for dense graphs. The existence of efficient approximation algorithms is then a consequence of the regularity lemma, as shown by Frieze and Kannan.

Applying our method to the Max Cut problem, we devise an algorithm that is faster than all previous algorithms, and is easier to describe and analyze.

References

  1. [ABS10]
    Arora, S., Barak, B., Steurer, D.: Subexponential algorithms for unique games and related problems. In: FOCS, pp. 563–572 (2010)Google Scholar
  2. [AN06]
    Alon, N., Naor, A.: Approximating the cut-norm via grothendieck’s inequality. SIAM J. Comput. 35(4), 787–803 (2006)MathSciNetMATHCrossRefGoogle Scholar
  3. [BRS11]
    Barak, B., Raghavendra, P., Steurer, D.: Rounding semidefinite programming hierarchies via global correlation. In: FOCS, pp. 472–481 (2011)Google Scholar
  4. [CCF09]
    Coja-Oghlan, A., Cooper, C., Frieze, A.: An efficient sparse regularity concept. In: SODA, pp. 207–216. Society for Industrial and Applied Mathematics, Philadelphia (2009)Google Scholar
  5. [DKKV05]
    Fernandez De la Vega, W., Karpinski, M., Kannan, R., Vempala, S.: Tensor decomposition and approximation schemes for constraint satisfaction problems. In: STOC, pp. 747–754. ACM, New York (2005)Google Scholar
  6. [FK96]
    Frieze, A.M., Kannan, R.: The regularity lemma and approximation schemes for dense problems. In: FOCS, p. 12. IEEE Computer Society, Washington, DC (1996)Google Scholar
  7. [FK99]
    Frieze, A.M., Kannan, R.: Quick approximation to matrices and applications. Combinatorica 19(2), 175–220 (1999)MathSciNetMATHCrossRefGoogle Scholar
  8. [GS11]
    Guruswami, V., Sinop, A.K.: Lasserre hierarchy, higher eigenvalues, and approximation schemes for graph partitioning and quadratic integer programming with psd objectives. In: FOCS, pp. 482–491 (2011)Google Scholar
  9. [GS12]
    Guruswami, V., Sinop, A.K.: Faster sdp hierarchy solvers for local rounding algorithms. In: FOCS (2012)Google Scholar
  10. [GS13]
    Guruswami, V., Sinop, A.K.: Lasserre sdps, l1-embeddings, and approximating non-uniform sparsest cut via generalized spectra. In: SODA (2013)Google Scholar
  11. [GW95]
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115–1145 (1995)MathSciNetMATHCrossRefGoogle Scholar
  12. [Kol11]
    Kolla, A.: Spectral algorithms for unique games. Comput. Complex. 20(2), 177–206 (2011)MathSciNetMATHCrossRefGoogle Scholar
  13. [KT07]
    Kolla, A., Tulsiani, M.: Playing unique games using graph spectra. Manuscript (2007)Google Scholar
  14. [Sze78]
    Szemeredi, E.: Regular partitions of graphs. In: Problemes Combinatoires et Theorie des Graphes. Colloq. Internat. CNRS, vol. 260, pp. 399–401. CNRS, Paris (1978)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Shayan Oveis Gharan
    • 1
  • Luca Trevisan
    • 2
  1. 1.Department of Management Science and EngineeringStanford UniversityUSA
  2. 2.Department of Computer ScienceStanford UniversityUSA

Personalised recommendations