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Acta Informatica

, Volume 55, Issue 1, pp 57–80 | Cite as

Online edge coloring of paths and trees with a fixed number of colors

  • Lene M. Favrholdt
  • Jesper W. Mikkelsen
Original Article
  • 101 Downloads

Abstract

We study a version of online edge coloring, where the goal is to color as many edges as possible using only a given number, k, of available colors. All of our results are with regard to competitive analysis. Previous attempts to identify optimal algorithms for this problem have failed, even for bipartite graphs. Thus, in this paper, we analyze even more restricted graph classes, paths and trees. For paths, we consider \(k=2\), and for trees, we consider any \(k \ge 2\). We prove that a natural greedy algorithm called \({\textsc {First-Fit}}\) is optimal among deterministic algorithms, on paths as well as trees. For paths, we give a randomized algorithm, which is optimal and better than the best possible deterministic algorithm. For trees, we prove that to obtain a better competitive ratio than \({\textsc {First-Fit}}\), the algorithm would have to be both randomized and unfair (i.e., reject edges that could have been colored), and even such algorithms cannot be much better than \({\textsc {First-Fit}}\).

Notes

Acknowledgments

The authors would like to thank the anonymous reviewers for helpful comments on this work and its presentation.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdense MDenmark

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