Skip to main content
Springer Nature Link
Log in

Approximating the Max Edge-Coloring Problem

  • Conference paper
Combinatorial Algorithms (IWOCA 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5874))

Included in the following conference series:

  • 793 Accesses

Abstract

We study the weighted generalization of the edge coloring problem where the goal is to minimize the sum of the weights of the heaviest edges in the color classes. In particular, we deal with the approximability of this problem on bipartite graphs and trees. We first improve the best known approximation ratios for bipartite graphs of maximum degree \({\it \Delta} \geq 7\). For trees we present a polynomial 3/2-approximation algorithm, which is the first one for any special graph class with an approximation ratio less than the known ratio of two for general graphs. Also for trees, we propose a moderately exponential approximation algorithm that improves the 3/2 ratio with running time much better than that needed for the computation of an optimal solution.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Björklund, A., Husfeldt, T.: Inclusion–exclusion algorithms for counting set partitions. In: 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), pp. 575–582. IEEE Computer Society, Los Alamitos (2006)

    Chapter  Google Scholar 

  2. Cole, R., Ost, K., Schirra, S.: Edge-coloring bipartite multigraphs in O(E logD) time. Combinatorica 21, 5–12 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. de Werra, D., Demange, M., Escoffier, B., Monnot, J., Paschos, V.T.: Weighted coloring on planar, bipartite and split graphs: Complexity and approximation. Discrete Applied Mathematics 157, 819–832 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. de Werra, D., Hoffman, A.J., Mahadev, N.V.R., Peled, U.N.: Restrictions and preassignments in preemptive open shop scheduling. Discrete Applied Mathematics 68, 169–188 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Demange, M., de Werra, D., Monnot, J., Paschos, V.T.: Time slot scheduling of compatible jobs. Journal of Scheduling 10, 111–127 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Epstein, L., Levin, A.: On the max coloring problem. In: Kaklamanis, C., Skutella, M. (eds.) WAOA 2007. LNCS, vol. 4927, pp. 142–155. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  7. Escoffier, B., Monnot, J., Paschos, V.T.: Weighted coloring: further complexity and approximability results. Information Processing Letters 97, 98–103 (2006)

    MATH  MathSciNet  Google Scholar 

  8. Holyer, I.: The NP-completeness of edge-coloring. SIAM Journal on Computing 10, 718–720 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kesselman, A., Kogan, K.: Nonpreemptive scheduling of optical switches. IEEE Transactions on Communications 55, 1212–1219 (2007)

    Article  Google Scholar 

  10. König, D.: Über graphen und ihre anwendung auf determinantentheorie und mengenlehre. Mathematische Annalen 77, 453–465 (1916)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kubale, M.: Some results concerning the complexity of restricted colorings of graphs. Discrete Applied Mathematics 36, 35–46 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lucarelli, G., Milis, I., Paschos, V.T.: On the maximum edge coloring problem. In: Bampis, E., Skutella, M. (eds.) WAOA 2008. LNCS, vol. 5426, pp. 279–292. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  13. Lucarelli, G., Milis, I., Paschos, V.T.: On the max-weight edge coloring problem. Journal of Combinatorial Optimization (in press)

    Google Scholar 

  14. Micali, S., Vazirani, V.V.: An \({O(\sqrt{|V|}|E|)}\) algorithm for finding maximum matching in general graphs. In: 21st Annual IEEE Symposium on Foundations of Computer Science (FOCS 1980), pp. 17–27. IEEE Computer Society, Los Alamitos (1980)

    Chapter  Google Scholar 

  15. Pemmaraju, S.V., Raman, R.: Approximation algorithms for the max-coloring problem. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1064–1075. Springer, Heidelberg (2005)

    Google Scholar 

  16. Pemmaraju, S.V., Raman, R., Varadarajan, K.R.: Buffer minimization using max-coloring. In: 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2004), pp. 562–571 (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LAMSADE, Université Paris-Dauphine, France

    Nicolas Bourgeois & Vangelis Th. Paschos

  2. Dept. of Informatics, Athens University of Economics and Business, Greece

    Giorgio Lucarelli & Ioannis Milis

Authors
  1. Nicolas Bourgeois
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giorgio Lucarelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ioannis Milis
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vangelis Th. Paschos
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute for Theoretical Computer Science and Department of Applied Mathematics, Charles University, Prague, Czech Republic

    Jiří Fiala  & Jan Kratochvíl  & 

  2. School of Electrical Engineering and Computer Science, The University of Newcastle, University Drive, NSW 2308, Callaghan, Australia

    Mirka Miller

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bourgeois, N., Lucarelli, G., Milis, I., Paschos, V.T. (2009). Approximating the Max Edge-Coloring Problem. In: Fiala, J., Kratochvíl, J., Miller, M. (eds) Combinatorial Algorithms. IWOCA 2009. Lecture Notes in Computer Science, vol 5874. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10217-2_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-10217-2_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10216-5

  • Online ISBN: 978-3-642-10217-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics