Minimizing Average Completion of Dedicated Tasks and Interval Graphs

  • Magnús M. Halldórsson
  • Guy Kortsarz
  • Hadas Shachnai
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2129)


Scheduling dependent jobs on multiple machines is modeled as a graph (multi)coloring problem. The focus of this work is on the sum of completion times measure. This is known as the sum (multi)coloring of the conflict graph. We also initiate the study of the waiting time and the robust throughput of colorings. For uniform-length tasks we give an algorithm which simultaneously approximates these two measures, as well as sum coloring and the chromatic number, within constant factor, for any graph in which the k-colorable subgraph problem is polynomially solvable. In particular, this improves the best approximation ratio known for sum coloring interval graphs from 2 to 1.665.

We then consider the problem of scheduling non-preemptively tasks (of non-uniform lengths) that require exclusive use of dedicated processors. The objective is to minimize the sum of completion times. We obtain the first constant factor approximations for this problem, when each task uses a constant number of processors.


Completion Time Chromatic Number Interval Graph Chordal Graph Average Completion 
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  1. AB+00.
    F. Afrati, E. Bampis, A. Fishkin, K. Jansen, and C. Kenyon. Scheduling to minimize the average completion time of dedicated tasks. In FSTTCS 2000, LNCS, Delhi.Google Scholar
  2. BK98.
    A. Bar-Noy and G. Kortsarz. The minimum color-sum of bipartite graphs. Journal of Algorithms, 28:339–365, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  3. BBH+98.
    A. Bar-Noy, M. Bellare, M. M. Halldórsson, H. Shachnai, and T. Tamir. On chromatic sums and distributed resource allocation. Information and Computation, 140:183–202, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  4. BH94.
    D. Bullock and C. Hendrickson. Roadway traffic control software. IEEE Transactions on Control Systems Technology, 2:255–264, 1994.CrossRefGoogle Scholar
  5. BHK.
    A. Bar-Noy, M. M. Halldórsson, G. Kortsarz. Tight Bound for the Sum of a Greedy Coloring. Information Processing Letters, 1999.Google Scholar
  6. BHK+99.
    A. Bar-Noy, M. M. Halldórsson, G. Kortsarz, H. Shachnai, and R. Salman. Sum Multicoloring of Graphs. Journal of Algorithms, 37(2):422–450, November 2000.Google Scholar
  7. BKR96.
    P. Brucker and A. Krämer. Polynomial algorithms for resource-constrained and multiprocessor task scheduling problems. European Journal of Operational Research, 90:214–226, 1996.zbMATHCrossRefGoogle Scholar
  8. F80.
    A. Frank. On Chain and Antichain Families of a Partially Ordered Set. J. Combinatorial Theory, Series B, 29: 176–184, 1980.zbMATHCrossRefGoogle Scholar
  9. G01.
    M. Gonen, Coloring Problems on Interval Graphs and Trees. M.Sc. Thesis. School of Computer Science, The Open Univ., Israel, 2001.Google Scholar
  10. HK99.
    M. M. Halldórsson and G. Kortsarz. Multicoloring Planar Graphs and Partial k-trees. In Proceedings of the Second International Workshop on Approximation algorithms (APPROX’ 99). Lecture Notes in Computer Science Vol. 1671, Springer-Verlag, August 1999.Google Scholar
  11. HK+99.
    M. M. Halldórsson, G. Kortsarz, A. Proskurowski, R. Salman, H. Shachnai, and J. A. Telle. Multi-Coloring Trees. In Proceedings of the Fifth International Computing and Combinatorics Conference (COCOON), Tokyo, Japan, Lecture Notes in Computer Science Vol. 1627, Springer-Verlag, July 1999.Google Scholar
  12. J97.
    K. Jansen. The Optimum Cost Chromatic Partition Problem. Proc. of the Third Italian Conference on Algorithms and Complexity (CIAC’ 97). LNCS 1203, 1997.Google Scholar
  13. K89.
    E. Kubicka. The Chromatic Sum of a Graph. PhD thesis, Western Michigan University, 1989.Google Scholar
  14. K96.
    M. Kubale. Preemptive versus nonpreemptive scheduling of biprocessor tasks on dedicated processors. European Journal of Operational Research 94:242–251, 1996.zbMATHCrossRefGoogle Scholar
  15. NSS99.
    S. Nicoloso, M. Sarrafzadeh and X. Song. On the Sum Coloring Problem on Interval Graphs. Algorithmica, 23:109–126, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  16. W97.
    G. Woeginger. Private communication, 1997.Google Scholar
  17. YG87.
    M. Yannakakis and F. Gavril. The maximum k-colorable subgraph problem for chordal graphs. Inform. Proc. Letters, 24:133–137, 1987.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Magnús M. Halldórsson
    • 1
    • 2
  • Guy Kortsarz
    • 3
  • Hadas Shachnai
    • 4
  1. 1.Dept. of Computer ScienceUniversity of IcelandReykjavikIceland
  2. 2.Iceland Genomics Corp.Reykjavik
  3. 3.Dept. of Computer ScienceOpen UniversityRamat AvivIsrael
  4. 4.Dept. of Computer ScienceTechnion, HaifaIsrael

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