Towards a Study of Low-Complexity Graphs

  • Sanjeev Arora
  • David Steurer
  • Avi Wigderson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5555)

Abstract

We propose the study of graphs that are defined by low-complexity distributed and deterministic agents. We suggest that this viewpoint may help introduce the element of individual choice in models of large scale social networks. This viewpoint may also provide interesting new classes of graphs for which to design algorithms.

We focus largely on the case where the “low complexity” computation is AC0. We show that this is already a rich class of graphs that includes examples of lossless expanders and power-law graphs. We give evidence that even such low complexity graphs present a formidable challenge to algorithms designers. On the positive side, we show that many algorithms from property testing and data sketching can be adapted to give meaningful results for low-complexity graphs.

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References

  1. 1.
    Mitzenmacher, M.: A brief history of generative models for power law and lognormal distributions. Internet Mathematics 1(2) (2003)Google Scholar
  2. 2.
    Hopcroft, J.: Personal communication (2008)Google Scholar
  3. 3.
    Galperin, H., Wigderson, A.: Succinct representations of graphs. Inf. Control 56(3), 183–198 (1983)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Papadimitriou, C.H.: On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48(3), 498–532 (1994)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Razborov, A.A., Rudich, S.: Natural proofs. J. Comput. Syst. Sci. 55(1), 24–35 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Arora, S., Barak, B.: Computational Complexity: A modern approach. Cambridge University Press, Cambridge (2009)CrossRefMATHGoogle Scholar
  7. 7.
    Nickel, C.L.M.: Random dot product graphs: A model for social networks. PhD thesis, Johns Hopkins University (2008)Google Scholar
  8. 8.
    Young, S.J., Scheinerman, E.R.: Random dot product graph models for social networks. In: Bonato, A., Chung, F.R.K. (eds.) WAW 2007. LNCS, vol. 4863, pp. 138–149. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Ajtai, M.: \(\sigma_1^1\) formulae on finite structures. Annals of Pure Appl. Logic 24 (1983)Google Scholar
  10. 10.
    Furst, M.L., Saxe, J.B., Sipser, M.: Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory 17(1), 13–27 (1984)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Viola, E.: On approximate majority and probabilistic time. In: IEEE Conference on Computational Complexity, pp. 155–168 (2007)Google Scholar
  12. 12.
    Hastad, J.: Almost optimal lower bounds for small depth circuits. In: Randomness and Computation, pp. 6–20. JAI Press (1989)Google Scholar
  13. 13.
    Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. AMS (43), 439–561 (2006)Google Scholar
  14. 14.
    Capalbo, M.R., Reingold, O., Vadhan, S.P., Wigderson, A.: Randomness conductors and constant-degree lossless expanders. In: STOC, pp. 659–668 (2002)Google Scholar
  15. 15.
    Alon, N., Schwartz, O., Shapira, A.: An elementary construction of constant-degree expanders. In: SODA, pp. 454–458 (2007)Google Scholar
  16. 16.
    Reingold, O., Vadhan, S.P., Wigderson, A.: Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. In: FOCS, pp. 3–13 (2000)Google Scholar
  17. 17.
    Mihail, M., Papadimitriou, C.H.: On the eigenvalue power law. In: Rolim, J.D.P., Vadhan, S.P. (eds.) RANDOM 2002. LNCS, vol. 2483, pp. 254–262. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  18. 18.
    Babai, L., Fortnow, L., Lund, L.: Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity 1, 3–40 (1991); Prelim version FOCS 1990Google Scholar
  19. 19.
    Arora, S., Safra, S.: Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM 45(1), 70–122 (1998); Prelim version FOCS 1992Google Scholar
  20. 20.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. Journal of the ACM 45(3), 501–555 (1998); Prelim version FOCS 1992Google Scholar
  21. 21.
    Feige, U., Goldwasser, S., Lovász, L., Safra, S., Szegedy, M.: Interactive proofs and the hardness of approximating cliques. Journal of the ACM 43(2), 268–292 (1996); Prelim version FOCS 1991Google Scholar
  22. 22.
    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)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. J. ACM 45(4), 653–750 (1998)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Alon, N., de la Vega, W.F., Kannan, R., Karpinski, M.: Random sampling and approximation of max-csps. J. Comput. Syst. Sci. 67(2), 212–243 (2003)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Alon, N., Fischer, E., Newman, I., Shapira, A.: A combinatorial characterization of the testable graph properties: it’s all about regularity. In: STOC, pp. 251–260 (2006)Google Scholar
  26. 26.
    Benjamini, I., Schramm, O., Shapira, A.: Every minor-closed property of sparse graphs is testable. In: STOC, pp. 393–402 (2008)Google Scholar
  27. 27.
    Frieze, A.M., Kannan, R., Vempala, S.: Fast monte-carlo algorithms for finding low-rank approximations. In: FOCS, pp. 370–378 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Sanjeev Arora
    • 1
  • David Steurer
    • 1
  • Avi Wigderson
    • 2
  1. 1.Computer Science Dept.Princeton UniversityPrincetonUSA
  2. 2.Institute for Advanced StudyPrincetonUSA

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