Bounded Max-colorings of Graphs

  • Evripidis Bampis
  • Alexander Kononov
  • Giorgio Lucarelli
  • Ioannis Milis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6506)

Abstract

In a bounded max-coloring of a vertex/edge weighted graph, each color class is of cardinality at most b and of weight equal to the weight of the heaviest vertex/edge in this class. The bounded max-vertex/edge-coloring problems ask for such a coloring minimizing the sum of all color classes’ weights. These problems generalize the well known max-coloring problems by taking into account the number of available resources (colors) in practical applications. In this paper we present complexity results and approximation algorithms for the bounded max-coloring problems on general graphs, bipartite graphs and trees.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N.: A note on the decomposition of graphs into isomorphic matchings. Acta Mathematica Hungarica 42, 221–223 (1983)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Baker, B.S., Coffman Jr., E.G.: Mutual exclusion scheduling. Theoretical Computer Science 162, 225–243 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bodlaender, H.L., Jansen, K.: Restrictions of graph partition problems. Part I. Theoretical Computer Science 148, 93–109 (1995)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bongiovanni, G., Coppersmith, D., Wong, C.K.: An optimum time slot assignment algorithm for an SS/TDMA system with variable number of transponders. IEEE Trans. on Communications 29, 721–726 (1981)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Boudhar, M., Finke, G.: Scheduling on a batch machine with job compatibilities. Belgian Journal of Oper. Res., Statistics and Computer Science 40, 69–80 (2000)MathSciNetMATHGoogle Scholar
  6. 6.
    Bourgeois, N., Lucarelli, G., Milis, I., Paschos, V.T.: Approximating the max-edge-coloring problem. Theoretical Computer Science 411, 3055–3067 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chvátal, V.: A greedy heuristic for the set-covering problem. Mathematics of Operations Research 4, 233–235 (1979)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    de Werra, D., Hertz, A., Kobler, D., Mahadev, N.V.R.: Feasible edge coloring of trees with cardinality constraints. Discrete Mathematics 222, 61–72 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Demange, M., de Werra, D., Monnot, J., Paschos, V.T.: Time slot scheduling of compatible jobs. Journal of Scheduling 10, 111–127 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Dror, M., Finke, G., Gravier, S., Kubiak, W.: On the complexity of a restricted list-coloring problem. Discrete Mathematics 195, 103–109 (1999)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    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)CrossRefGoogle Scholar
  13. 13.
    Escoffier, B., Monnot, J., Paschos, V.T.: Weighted coloring: Further complexity and approximability results. Information Processing Letters 97, 98–103 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gardi, F.: Mutual exclusion scheduling with interval graphs or related classes. Part II. Discrete Applied Mathematics 156, 794–812 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gravier, S., Kobler, D., Kubiak, W.: Complexity of list coloring problems with a fixed total number of colors. Discrete Applied Mathematics 117, 65–79 (2002)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Holyer, I.: The NP-completeness of edge-coloring. SIAM Journal on Computing 10, 718–720 (1981)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Jarvis, M., Zhou, B.: Bounded vertex coloring of trees. Discrete Mathematics 232, 145–151 (2001)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kavitha, T., Mestre, J.: Max-coloring paths: Tight bounds and extensions. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 87–96. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  19. 19.
    Kesselman, A., Kogan, K.: Nonpreemptive scheduling of optical switches. IEEE Trans. on Communications 55, 1212–1219 (2007)CrossRefGoogle Scholar
  20. 20.
    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)CrossRefGoogle Scholar
  21. 21.
    Pemmaraju, S.V., Raman, R., Varadarajan, K.R.: Buffer minimization using max-coloring. In: SODA 2004, pp. 562–571 (2004)Google Scholar
  22. 22.
    Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: STOC 2006, pp. 681–690 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Evripidis Bampis
    • 1
  • Alexander Kononov
    • 2
  • Giorgio Lucarelli
    • 3
  • Ioannis Milis
    • 4
  1. 1.LIP6Université Pierre et Marie CurieFrance
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia
  3. 3.LAMSADEUniversité Paris-Dauphine and CNRS FRE 3234France
  4. 4.Dept. of InformaticsAthens University of Economics and BusinessGreece

Personalised recommendations