Online Dual Edge Coloring of Paths and Trees

  • Lene M. Favrholdt
  • Jesper W. Mikkelsen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8952)


We study a dual 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. For paths, we consider \(k=2\), and for trees, we consider any \(k \ge 2\). We prove that a natural greedy algorithm called First-Fit is optimal among deterministic algorithms on paths as well as trees. This is the first time that an optimal algorithm for online dual edge coloring has been identified for a class of graphs. For paths, we give a randomized algorithm, which is optimal and better than the best possible deterministic algorithm. Again, it is the first time that this has been done for a class of graphs. For trees, we also show that even randomized algorithms cannot be much better than First-Fit.


  1. 1.
    Bar-Noy, A., Motwani, R., Naor, J.: The greedy algorithm is optimal for on-line edge coloring. Inf. Process. Lett. 44(5), 251–253 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  3. 3.
    Boyar, J., Favrholdt, L.M.: The relative worst order ratio for online algorithms. ACM Trans. Algorithms 3(2), 22 (2007)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Boyar, J., Favrholdt, L.M., Larsen, K.S.: The relative worst-order ratio applied to paging. J. Comput. Syst. Sci. 73, 818–843 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Chen, Z.-Z., Konno, S., Matsushita, Y.: Approximating maximum edge 2-coloring in simple graphs. Discret. Appl. Math. 158(17), 1894–1901 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Ehmsen, M.R., Favrholdt, L.M., Kohrt, J.S., Mihai, R.: Comparing first-fit and next-fit for online edge coloring. Theor. Comput. Sci. 411(16–18), 1734–1741 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Favrholdt, L.M., Mikkelsen, J.W.: Online dual edge coloring of paths and trees. ArXiv e-prints, 1405.3817 [cs.DS] (2014)Google Scholar
  8. 8.
    Favrholdt, L.M., Nielsen, M.N.: On-line edge-coloring with a fixed number of colors. Algorithmica 35(2), 176–191 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Feige, U., Ofek, E., Wieder, U.: Approximating maximum edge coloring in multigraphs. In: Jansen, K., Leonardi, S., Vazirani, V.V. (eds.) APPROX 2002. LNCS, vol. 2462, pp. 108–121. Springer, Heidelberg (2002) CrossRefGoogle Scholar
  10. 10.
    Kamiński, M., Kowalik, Ł.: Approximating the maximum 3- and 4-edge-colorable subgraph. In: Kaplan, H. (ed.) SWAT 2010. LNCS, vol. 6139, pp. 395–407. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  11. 11.
    Karlin, A.R., Manasse, M.S., Rudolph, L., Sleator, D.D.: Competitive snoopy caching. Algorithmica 3, 77–119 (1988)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Kierstead, H.A.: Coloring graphs on-line. In: Fiat, A., Woeginger, G.J. (eds.) Online Algorithms 1996. LNCS, vol. 1442, pp. 281–305. Springer, Heidelberg (1998) CrossRefGoogle Scholar
  13. 13.
    Kosowski, A.: Approximating the maximum 2- and 3-edge-colorable subgraph problems. Discret. Appl. Math. 157(17), 3593–3600 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Rizzi, R.: Approximating the maximum 3-edge-colorable subgraph problem. Discret. Math. 309(12), 4166–4170 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Commun. ACM 28(2), 202–208 (1985)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Yao, A.C-C.: Probabilistic computations: Toward a unified measure of complexity (extended abstract). In: FOCS, pp. 222–227 (1977)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdenseDenmark

Personalised recommendations