On the Locality of Some NP-Complete Problems

  • Leonid Barenboim
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7392)

Abstract

We consider the distributed message-passing \({\cal LOCAL}\) model. In this model a communication network is represented by a graph where vertices host processors, and communication is performed over the edges. Computation proceeds in synchronous rounds. The running time of an algorithm is the number of rounds from the beginning until all vertices terminate. Local computation is free. An algorithm is called local if it terminates within a constant number of rounds. The question of what problems can be computed locally was raised by Naor and Stockmeyer [16] in their seminal paper in STOC’93. Since then the quest for problems with local algorithms, and for problems that cannot be computed locally, has become a central research direction in the field of distributed algorithms [9,11,13,17].

We devise the first local algorithm for an NP-complete problem. Specifically, our randomized algorithm computes, with high probability, an O(n 1/2 + ε ·χ)-coloring within O(1) rounds, where ε > 0 is an arbitrarily small constant, and χ is the chromatic number of the input graph. (This problem was shown to be NP-complete in [21].) On our way to this result we devise a constant-time algorithm for computing (O(1), O(n 1/2 + ε ))-network-decompositions. Network-decompositions were introduced by Awerbuch et al. [1], and are very useful for solving various distributed problems. The best previously-known algorithm for network-decomposition has a polylogarithmic running time (but is applicable for a wider range of parameters) [15]. We also devise a Δ1 + ε -coloring algorithm for graphs with sufficiently large maximum degree Δ that runs within O(1) rounds. It improves the best previously-known result for this family of graphs, which is O(log* n) [19].

Keywords

Maximum Degree Chromatic Number Local Algorithm Legal Coloring Input Graph 
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.
    Awerbuch, B., Goldberg, A.V., Luby, M., Plotkin, S.: Network decomposition and locality in distributed computation. In: Proc. 30th IEEE Symp. on Foundations of Computer Science, pp. 364–369 (October 1989)Google Scholar
  2. 2.
    Barenboim, L.: On the locality of some NP-complete problems (2012), http://arXiv.org/abs/1204.6675
  3. 3.
    Barenboim, L., Elkin, M.: Sublogarithmic distributed MIS algorithm for sparse graphs using Nash-Williams decomposition. In: Proc. 27th ACM Symp. on Principles of Distributed Computing, pp. 25–34 (2008)Google Scholar
  4. 4.
    Barenboim, L., Elkin, M.: Distributed (Δ + 1)-coloring in linear (in Δ) time. In: Proc. 41st ACM Symp. on Theory of Computing, pp. 111–120 (2009)Google Scholar
  5. 5.
    Barenboim, L., Elkin, M.: Deterministic distributed vertex coloring in polylogarithmic time. In: Proc. 29th ACM Symp. on Principles of Distributed Computing, pp. 410–419 (2010)Google Scholar
  6. 6.
    Cole, R., Vishkin, U.: Deterministic coin tossing with applications to optimal parallel list ranking. Information and Control 70(1), 32–53 (1986)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Goldberg, A., Plotkin, S.: Efficient parallel algorithms for (Δ + 1) -coloring and maximal independent set problem. In: Proc. 19th ACM Symp. on Theory of Computing, pp. 315–324 (1987)Google Scholar
  8. 8.
    Fraigniaud, P., Korman, A., Peleg, D.: Local Distributed Decision. In: Proc. 52nd IEEE Symp. on Foundations of Computer Science, pp. 708–717 (2011)Google Scholar
  9. 9.
    Kuhn, F.: Weak graph colorings: distributed algorithms and applications. In: Proc. 21st ACM Symp. on Parallel Algorithms and Architectures, pp. 138–144 (2009)Google Scholar
  10. 10.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: The price of being near-sighted. In: Proc. 17th ACM-SIAM Symp. on Discrete Algorithms, pp. 980–989 (2006)Google Scholar
  11. 11.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: Local Computation: Lower and Upper Bounds (2010), http://arXiv.org/abs/1011.5470
  12. 12.
    Kuhn, F., Wattenhofer, R.: Constant-time distributed dominating set approximation. In: 22nd ACM Symp. Principles of Distributed Computing, pp. 25–32 (2003)Google Scholar
  13. 13.
    Lenzen, C., Oswald, Y., Wattenhofer, R.: What Can Be Approximated Locally? Case Study: Dominating Sets in Planar Graphs. In: Proc. 20th ACM Symp. on Parallelism in Algorithms and Architectures, pp. 46–54 (2008); See also TIK report number 331, ETH Zurich (2010)Google Scholar
  14. 14.
    Linial, N.: Distributive Graph Algorithms-Global Solutions from Local Data. In: Proc. 28th IEEE Symp. on Foundations of Computer Science, pp. 331–335 (1987)Google Scholar
  15. 15.
    Linial, N., Saks, M.: Low diameter graph decompositions. Combinatorica 13(4), 441–454 (1993)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Naor, M., Stockmeyer, L.: What can be computed locally? In: Proc. 25th ACM Symp. on Theory of Computing, pp. 184–193 (1993)Google Scholar
  17. 17.
    Panconesi, A., Rizzi, R.: Some simple distributed algorithms for sparse networks. Distributed Computing 14(2), 97–100 (2001)CrossRefGoogle Scholar
  18. 18.
    Panconesi, A., Srinivasan, A.: On the complexity of distributed network decomposition. Journal of Algorithms 20(2), 581–592 (1995)MathSciNetGoogle Scholar
  19. 19.
    Schneider, J., Wattenhofer, R.: A New Technique For Distributed Symmetry Breaking. In: 29th ACM Symp. Principles of Distributed Computing, pp. 257–266 (2010)Google Scholar
  20. 20.
    Schneider, J., Wattenhofer, R.: Distributed Coloring Depending on the Chromatic Number or the Neighborhood Growth. In: Kosowski, A., Yamashita, M. (eds.) SIROCCO 2011. LNCS, vol. 6796, pp. 246–257. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  21. 21.
    Zuckerman, D.: Linear Degree Extractors and the Inapproximability of Max Clique and Chromatic Number. Theory of Computing 3(1), 103–128 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Leonid Barenboim
    • 1
  1. 1.Department of Computer ScienceBen-Gurion University of the NegevBeer-ShevaIsrael

Personalised recommendations