Advertisement

Online Coloring of Bipartite Graphs with and without Advice

  • Maria Paola Bianchi
  • Hans-Joachim Böckenhauer
  • Juraj Hromkovič
  • Lucia Keller
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)

Abstract

In the online version of the well-known graph coloring problem, the vertices appear one after the other together with the edges to the already known vertices and have to be irrevocably colored immediately after their appearance. We consider this problem on bipartite, i.e., two-colorable graphs. We prove that 1.13747·log2 n colors are necessary for any deterministic online algorithm to color any bipartite graph on n vertices, thus improving on the previously known lower bound of log2 n + 1 for sufficiently large n.

Recently, the advice complexity was introduced as a method for a fine-grained analysis of the hardness of online problems. We apply this method to the online coloring problem and prove (almost) tight linear upper and lower bounds on the advice complexity of coloring a bipartite graph online optimally or using 3 colors. Moreover, we prove that \(O(\sqrt{n})\) advice bits are sufficient for coloring any graph on n vertices with at most ⌈log2 n⌉ colors.

Keywords

Bipartite Graph Competitive Ratio Online Algorithm Additional Vertex Common Color 
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.
    Bean, D.R.: Effective Coloration. J. Symbolic Logic 41(2), 469–480 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Böckenhauer, H.-J., Komm, D., Královič, R., Rossmanith, P.: On the Advice Complexity of the Knapsack Problem. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 61–72. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  3. 3.
    Böckenhauer, H.-J., Komm, D., Královič, R., Královič, R., Mömke, T.: On the Advice Complexity of Online Problems. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 331–340. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press (1998)Google Scholar
  5. 5.
    Dobrev, S., Královič, R., Pardubská, D.: How Much Information about the Future Is Needed? In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds.) SOFSEM 2008. LNCS, vol. 4910, pp. 247–258. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Emek, Y., Fraigniaud, P., Korman, A., Rosén, A.: Online Computation with Advice. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 427–438. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Finch, S.R.: Mathematical Constants (Encyclopedia of Mathematics and its Applications). Cambridge University Press, New York (2003)Google Scholar
  8. 8.
    Forišek, M., Keller, L., Steinová, M.: Advice Complexity of Online Coloring for Paths. In: Dediu, A.-H., Martín-Vide, C. (eds.) LATA 2012. LNCS, vol. 7183, pp. 228–239. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  9. 9.
    Gyárfás, A., Lehel, J.: On-line and first fit colorings of graphs. Journal of Graph Theory 12(2), 217–227 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Hromkovič, J., Královič, R., Královič, R.: Information Complexity of Online Problems. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 24–36. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Kierstead, H.A.: Recursive and on-line graph coloring. In: Ershov, Y.L., Goncharov, S.S., Nerode, A., Remmel, J.B., Marek, V.W., (eds.) Handbook of Recursive Mathematics Volume 2: Recursive Algebra, Analysis and Combinatorics. Studies in Logic and the Foundations of Mathematics, vol. 139, pp. 1233–1269. Elsevier (1998)Google Scholar
  12. 12.
    Kierstead, H.A., Trotter, W.T.: On-line graph coloring. In: McGeoch, L.A., Sleator, D.D. (eds.) On-line Algorithms. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 7, pp. 85–92. AMS|DIMACS|ACM (1992)Google Scholar
  13. 13.
    Lovász, L., Saks, M.E., Trotter, W.T.: An on-line graph coloring algorithm with sublinear performance ratio. Discrete Mathematics 75(1–3), 319–325 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Sloane, N.J.A.: Sequence A000073 in The On-Line Encyclopedia of Integer Sequences (2012), Published electronically, http://oeis.org/A000073
  15. 15.
    Vishwanathan, S.: Randomized online graph coloring. Journal of Algorithms 13(4), 657–669 (1992)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Maria Paola Bianchi
    • 1
  • Hans-Joachim Böckenhauer
    • 2
  • Juraj Hromkovič
    • 2
  • Lucia Keller
    • 2
  1. 1.Dipartimento di InformaticaUniversità degli Studi di MilanoItaly
  2. 2.Department of Computer ScienceETH ZurichSwitzerland

Personalised recommendations