Advertisement

A Deterministic Algorithm for the Frieze-Kannan Regularity Lemma

  • Domingos Dellamonica
  • Subrahmanyam Kalyanasundaram
  • Daniel Martin
  • Vojtěch Rödl
  • Asaf Shapira
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6845)

Abstract

The Frieze-Kannan regularity lemma is a powerful tool in combinatorics. It has also found applications in the design of approximation algorithms and recently in the design of fast combinatorial algorithms for boolean matrix multiplication. The algorithmic applications of this lemma require one to efficiently construct a partition satisfying the conditions of the lemma.

Williams [24] recently asked if one can construct a partition satisfying the conditions of the Frieze-Kannan regularity lemma in deterministic sub-cubic time. We resolve this problem by designing an \(\tilde O(n^{\omega})\) time algorithm for constructing such a partition, where ω < 2.376 is the exponent of fast matrix multiplication. The algorithm relies on a spectral characterization of vertex partitions satisfying the properties of the Frieze-Kannan regularity lemma.

Keywords

Deterministic Algorithm Spectral Condition Deterministic Time Regularity Lemma Regular Partition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N.: Eigenvalues and expanders. Combinatorica 6, 83–96 (1986), doi:10.1007/BF02579166MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alon, N., Duke, R.A., Lefmann, H., Rödl, V., Yuster, R.: The algorithmic aspects of the regularity lemma. J. Algorithms 16, 80–109 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alon, N., Naor, A.: Approximating the cut-norm via Grothendieck’s inequality. In: Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, STOC 1984, pp. 72–80. ACM, New York (2004)CrossRefGoogle Scholar
  4. 4.
    Bansal, N., Williams, R.: Regularity lemmas and combinatorial algorithms. In: Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, pp. 745–754. IEEE Computer Society, Washington, DC, USA (2009)CrossRefGoogle Scholar
  5. 5.
    Bhatia, R.: Matrix Analysis. Springer, New York (1997)CrossRefzbMATHGoogle Scholar
  6. 6.
    Coppersmith, D.: Rapid multiplication of rectangular matrices. SIAM J. Computing 11, 467–471 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. In: Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, STOC 1987, pp. 1–6. ACM, New York (1987)Google Scholar
  8. 8.
    Duke, R.A., Lefmann, H., Rödl, V.: A fast approximation algorithm for computing the frequencies of subgraphs in a given graph. SIAM J. Comput. 24(3), 598–620 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Frieze, A., Kannan, R.: The regularity lemma and approximation schemes for dense problems. In: Annual IEEE Symposium on Foundations of Computer Science, p. 12 (1996)Google Scholar
  10. 10.
    Frieze, A., Kannan, R.: Quick approximation to matrices and applications. Combinatorica 19, 175–220 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Frieze, A., Kannan, R.: A simple algorithm for constructing Szemerédi’s regularity partition. Electr. J. Comb. 6 (1999) (electronic)Google Scholar
  12. 12.
    Gowers, W.T.: Quasirandomness, counting and regularity for 3-uniform hypergraphs. Comb. Probab. Comput. 15, 143–184 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gowers, W.T.: Lower bounds of tower type for Szemerédi’s uniformity lemma. Geometric And Functional Analysis 7, 322–337 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kohayakawa, Y., Rödl, V., Thoma, L.: An optimal algorithm for checking regularity. SIAM J. Comput. 32, 1210–1235 (2003); Earlier verison in SODA 2002MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Komlós, J., Shokoufandeh, A., Simonovits, M., Szemerédi, E.: The regularity lemma and its applications in graph theory, pp. 84–112. Springer-Verlag New York, Inc., New York (2002)zbMATHGoogle Scholar
  16. 16.
    Kuczyński, J., Woźniakowski, H.: Estimating the largest eigenvalue by the power and Lanczos algorithms with a random start. SIAM Journal on Matrix Analysis and Applications 13(4), 1094–1122 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lovász, L.: Very large graphs. In: Jerison, D., Mazur, B., Mrowka, T., Schmid, W., Stanley, R., Yau, S.T. (eds.) Current Developments in Mathematics (2008)Google Scholar
  18. 18.
    O’Leary, D.P., Stewart, G.W., Vandergraft, J.S.: Quasirandomness, counting and regularity for 3-uniform hypergraphs. Mathematics of Computation 33, 1289–1292 (1979)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Rödl, V., Schacht., M.: Regularity lemmas for graphs. In: Fete of Combinatorics and Computer Science. Bolyai Society Mathematical Studies, vol. 20, pp. 287–325. Springer, HeidelbergGoogle Scholar
  20. 20.
    Szemerédi, E.: On sets of integers containing no k elements in arithmetic progressions. Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica 27, 199–245 (1975)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Szemerédi, E.: Regular partitions of graphs. In: Problémes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), pp. 399–401. Éditions du Centre National de la Recherche Scientifique (CNRS), Paris (1978)Google Scholar
  22. 22.
    Trevisan, L.: Pseudorandomness in computer science and in additive combinatorics. In: An Irregular Mind. Bolyai Society Mathematical Studies, vol. 21, pp. 619–650. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  23. 23.
    Trevisan, L.: Lecture notes, http://lucatrevisan.wordpress.com/
  24. 24.
    Williams, R.: Private Communication (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Domingos Dellamonica
    • 1
  • Subrahmanyam Kalyanasundaram
    • 2
  • Daniel Martin
    • 3
  • Vojtěch Rödl
    • 1
  • Asaf Shapira
    • 4
  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA
  2. 2.School of Computer ScienceGeorgia Institute of TechnologyAtlantaUSA
  3. 3.Center for Mathematics, Computer Science and CognitionUniversidade Federal do ABCSanto AndréBrazil
  4. 4.School of Mathematics and School of Computer ScienceGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations