Sum-Multicoloring on Paths

  • Annamária Kovács
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2996)


The question whether the preemptive Sum Multicoloring (pSMC) problem is hard on paths was raised by Halldórsson et al. in [8]. The pSMC problem is a scheduling problem where the pairwise conflicting jobs are represented by a conflict graph, and the time lengths of jobs by integer weights on the nodes. The goal is to schedule the jobs so that the sum of their finishing times is minimized. In the paper we give an \(\mathcal{O}(n^3p)\) time algorithm for the pSMC problem on paths, where n is the number of nodes and p is the largest time length. The result easily carries over to cycles.


Local Minimum Schedule Problem Optimal Schedule Conflict Graph Valid Schedule 
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  1. 1.
    Bar-Noy, A., Kortsarz, G.: The minimum color-sum of bipartite graphs. Journal of Algorithms 28, 339–365 (1998)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Halldórsson, M.M., Kortsarz, G.: Tools for multicoloring with applications to planar graphs and partial k-trees. Journal of Algorithms 42(2), 334–366 (2002)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Kovács, A.: Sum-multicoloring on paths. Research Report MPI-I-2003-1-015, Max- Planck-Institut für Informatik (July 2003)Google Scholar
  4. 4.
    Kubicka, E.: The Chromatic Sum of a Graph. PhD thesis, Western Michigan University (1989)Google Scholar
  5. 5.
    Marx, D.: The complexity of tree multicolorings. In: Proc. 27th Intl. Symp. Math. Found. Comput. Sci (MFCS). LNCS, Springer, Heidelberg (2002)Google Scholar
  6. 6.
    Bar-Noy, A., Bellare, M., Halldórsson, M.M., Shachnai, H., Tamir, T.: On chromatic sums and distributed resource allocation. Inf. and Comput. 140, 183–202 (1998)MATHCrossRefGoogle Scholar
  7. 7.
    Bar-Noy, A., Halldórsson, M.M., Kortsarz, G., Shachnai, H., Salman, R.: Sum multicoloring of graphs. Journal of Algorithms 37, 422–450 (2000)MATHCrossRefGoogle Scholar
  8. 8.
    Halldórsson, M.M., Kortsarz, G., Proskurowski, A., Salman, R., Shachnai, H., Telle, J.A.: Multi-coloring trees. Information and Computation 180(2), 113–129 (2002)CrossRefGoogle Scholar
  9. 9.
    Szkaliczki, T.: Routing with minimum wire length in the dogleg-free Manhattan model is NP-complete. SIAM Journal on Computing 29(1), 274–287 (1999)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Annamária Kovács
    • 1
  1. 1.Max-Planck Institut für InformatikSaarbrückenGermany

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