Advertisement

Coloring Sparse Random k-Colorable Graphs in Polynomial Expected Time

  • Julia Böttcher
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3618)

Abstract

Feige and Kilian [5] showed that finding reasonable approximative solutions to the coloring problem on graphs is hard. This motivates the quest for algorithms that either solve the problem in most but not all cases, but are of polynomial time complexity, or that give a correct solution on all input graphs while guaranteeing a polynomial running time on average only. An algorithm of the first kind was suggested by Alon and Kahale in [1] for the following type of random k-colorable graphs: Construct a graph \(\mathcal{G}_{n,p,k}\) on vertex set V of cardinality n by first partitioning V into k equally sized sets and then adding each edge between these sets with probability p independently from each other. Alon and Kahale showed that graphs from \(\mathcal{G}_{n,p,k}\) can be k-colored in polynomial time with high probability as long as pc/n for some sufficiently large constant c. In this paper, we construct an algorithm with polynomial expected running time for k = 3 on the same type of graphs and for the same range of p. To obtain this result we modify the ideas developed by Alon and Kahale and combine them with techniques from semidefinite programming. The calculations carry over to general k.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Kahale, N.: A spectral technique for coloring random 3-colorable graphs. SIAM Journal on Computing 26(6), 1733–1748 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Coja-Oghlan, A.: Coloring semirandom graphs optimally. In: Proceedings of the 31st International Colloquium on Automata, Languages and Programming, pp. 383–395 (2004)Google Scholar
  3. 3.
    Coja-Oghlan, A., Moore, C., Sanwalani, V.: MAX k-CUT and approximating the chromatic number of random graphs. In: Proceedings of the 30th International Colloquium on Automata, Languages and Programming, pp. 200–211 (2003)Google Scholar
  4. 4.
    Dyer, M.E., Frieze, A.M.: The solution of some random NP-hard problems in polynomial expected time. Journal of Algorithms 10, 451–489 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Feige, U., Kilian, J.: Zero knowledge and the chromatic number. Journal of Computer and System Sciences 57(2), 187–199 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Frieze, A.M., Jerrum, M.: Improved approximation algorithms for MAX k-CUT and MAX BISECTION. Algorithmica 18, 61–77 (1997)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. W.H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  8. 8.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1993)zbMATHGoogle Scholar
  9. 9.
    Khanna, S., Linial, N., Safra, S.: On the hardness of approximating the chromatic number. Combinatorica 20(3), 393–415 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Krivelevich, M.: Deciding k-colorability in expected polynomial time. Information Processing Letters 81, 1–6 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kučera, L.: Expected behavior of graph colouring algorithms. In: Proceedings of the 1977 International Conference on Fundamentals of Computation Theory, pp. 447–451 (1977)Google Scholar
  12. 12.
    Subramanian, C.R.: Algorithms for coloring random k-colorable graphs. Combinatorics, Probability and Computing 9, 45–77 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Turner, J.S.: Almost all k-colorable graphs are easy to color. Journal of Algorithms 9, 253–261 (1988)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Julia Böttcher
    • 1
  1. 1.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany

Personalised recommendations