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Online Coloring of Bipartite Graphs with and without Advice

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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 at least ⌊1.13746⋅log2(n)−0.49887⌋ colors are necessary for any deterministic online algorithm to be able to color any given 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 bipartite graph on n vertices with at most ⌈log2 n⌉ colors.

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References

  1. Bean, D.R.: Effective coloration. J. Symb. Log. 41(2), 469–480 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  2. Böckenhauer, H.-J., Komm, D., Královič, R., Rossmanith, P.: On the advice complexity of the knapsack problem. In: Proc. of the 10th Latin American Symposium on Theoretical Informatics (LATIN 2012). LNCS, vol. 7256, pp. 61–72. Springer, Berlin (2012)

    Chapter  Google Scholar 

  3. Böckenhauer, H.-J., Komm, D., Královič, R., Královič, R., Mömke, T.: On the advice complexity of online problems. In: Proc. of the 20th International Symposium on Algorithms and Computation (ISAAC 2009). LNCS, vol. 5878, pp. 331–340. Springer, Berlin (2009)

    Google Scholar 

  4. Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  5. Dobrev, S., Královič, R., Pardubská, D.: How much information about the future is needed? In: Proc. of the 34th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2008). LNCS, vol. 4910, pp. 247–258. Springer, Berlin (2008)

    Chapter  Google Scholar 

  6. Elias, P.: Universal codeword sets and representations of the integers. IEEE Trans. Inf. Theory 21(2), 194–203 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  7. Emek, Y., Fraigniaud, P., Korman, A., Rosén, A.: Online computation with advice. In: Proc. of the 36th International Colloquium on Automata, Languages and Programming (ICALP 2009). LNCS, vol. 5555, pp. 427–438. Springer, Berlin (2009)

    Chapter  Google Scholar 

  8. Finch, S.R.: Mathematical Constants (Encyclopedia of Mathematics and Its Applications). Cambridge University Press, New York (2003)

    Google Scholar 

  9. Forišek, M., Keller, L., Steinová, M.: Advice complexity of online coloring for paths. In: Proc. of the 6th International Conference on Language and Automata Theory and Applications (LATA 2012). LNCS, vol. 7183, pp. 228–239. Springer, Berlin (2012)

    Chapter  Google Scholar 

  10. Gyárfás, A., Lehel, J.: On-line and first fit colorings of graphs. J. Graph Theory 12(2), 217–227 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hromkovič, J., Královič, R., Královič, R.: Information complexity of online problems. In: Proc. of the 35th International Symposium on Mathematical Foundations of Computer Science (MFCS 2010). LNCS, vol. 6281, pp. 24–36. Springer, Berlin (2010)

    Chapter  Google Scholar 

  12. 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, Vol. 2: Recursive Algebra, Analysis and Combinatorics. Studies in Logic and the Foundations of Mathematics, vol. 139, pp. 1233–1269. Elsevier, Amsterdam (1998)

    Google Scholar 

  13. 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 (1992). AMS/DIMACS/ACM

    Google Scholar 

  14. Komm, D., Královič, R.: Advice complexity and barely random algorithms. RAIRO ITA 45(2), 249–267 (2011)

    MATH  Google Scholar 

  15. Lovász, L., Saks, M.E., Trotter, W.T.: An on-line graph coloring algorithm with sublinear performance ratio. Discrete Math. 75(1–3), 319–325 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  16. Sloane, N.J.A.: Sequence A000073 in the on-line encyclopedia of integer sequences. Published electronically at http://oeis.org/A000073 (2012)

  17. Vishwanathan, S.: Randomized online graph coloring. J. Algorithms 13(4), 657–669 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees for helpful suggestions. The research was partially funded by the SNF grant 200021–141089.

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Authors and Affiliations

  1. Dipartimento di Informatica, Università degli Studi di Milano, Milan, Italy

    Maria Paola Bianchi

  2. Department of Computer Science, ETH Zurich, Zurich, Switzerland

    Hans-Joachim Böckenhauer, Juraj Hromkovič & Lucia Keller

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  1. Maria Paola Bianchi
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  2. Hans-Joachim Böckenhauer
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  3. Juraj Hromkovič
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  4. Lucia Keller
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Corresponding author

Correspondence to Hans-Joachim Böckenhauer.

Additional information

A preliminary version of this paper has been presented at the 18th Annual International Computing and Combinatorics Conference (COCOON 2012).

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Bianchi, M.P., Böckenhauer, HJ., Hromkovič, J. et al. Online Coloring of Bipartite Graphs with and without Advice. Algorithmica 70, 92–111 (2014). https://doi.org/10.1007/s00453-013-9819-7

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