Advertisement

Preemptive Scheduling on Dedicated Processors: Applications of Fractional Graph Coloring

  • Klaus Jansen
  • Lorant Porkolab
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1893)

Abstract

We study the problem of scheduling independent multiprocessor tasks, where for each task in addition to the processing time(s) there is a prespecified dedicated subset (or a family of alternative subsets) of processors which are required to process the task simultaneously. Focusing on problems where all required (alternative) subsets of processors have the same fixed cardinality, we present complexity results for computing preemptive schedules with minimum makespan closing the gap between computationally tractable and intractable instances. In particular, we show that for the dedicated version of the problem, optimal preemptive schedules of bi-processor tasks (i.e. tasks whose dedicated processor sets are all of cardinality two) can be computed in polynomial time. We give various extensions of this result including one to maximum lateness minimization with release times and due dates. All these results are based on a nice relation between preemptive scheduling and fractional coloring of graphs. In contrast to the positive results, we also prove that the problems of computing optimal preemptive schedules for three-processor tasks or for bi-processor tasks with (possible several) alternative modes are strongly NP-hard.

Keywords

Schedule Problem Polynomial Time Separation Problem Truth Assignment Preemptive Schedule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Bianco, J. Blazewicz, P. Dell’Olmo and M. Drozdowski, Scheduling preemptive multiprocessor tasks on dedicated processors, Performance Evaluation 20 (1994), 361–371.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    L. Bianco, J. Blazewicz, P. Dell’Olmo and M. Drozdowski, Preemptive multiprocessor task scheduling with release and time windows, Annals of Operations Research 70 (1997), 43–55.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. Blazewicz, M. Drozdowski, D. de Werra and J. Weglarz, Deadline scheduling of multiprocessor tasks, Discrete Applied Mathematics 65 (1996), 81–95.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    J. Blazewicz, K.H. Ecker, E. Pesch, G. Schmidt and J. Weglarz, Scheduling Computer and Manufacturing Processes, Springer Verlag, Berlin, 1996.zbMATHGoogle Scholar
  5. 5.
    M. Drozdowski, Scheduling multiprocessor tasks-an overview, European Journal on Operations Research, 94 (1996), 215–230.zbMATHCrossRefGoogle Scholar
  6. 6.
    U. Feige and J. Kilian, Zero knowledge and the chromatic number, Journal of Computer and System Sciences, 57 (1998), 187–199.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    M. Grötschel, L. Lovasz and A. Schrijver, The Ellipsoid Method and its consequences in combinatorial optimization, Combinatorica, 1 (1981) 169–197.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M. Grötschel, L. Lovasz and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer Verlag, Berlin, 1988.zbMATHGoogle Scholar
  9. 9.
    J.A. Hoogeveen, S.L. van de Velde and B. Veltman, Complexity of scheduling multiprocessor tasks with prespecified processor allocations, Discrete Applied Mathematics 55 (1994), 259–272.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    A. Krämer, Scheduling multiprocessor tasks on dedicated processors, Ph.D.thesis, Fachbereich Mathematik-Informatik, Universität Osnabrück, Germany, 1995.Google Scholar
  11. 11.
    H. Krawczyk and M. Kubale, An approximation algorithm for diagnostic test scheduling in multicomputer systems, IEEE Transactions on Computers 34 (1985), 869–872.CrossRefGoogle Scholar
  12. 12.
    M. Kubale, Preemptive versus nonpreemptive scheduling of biprocessor tasks on dedictated processors, European Journal of Operational Research, 94 (1996), 242–251.zbMATHCrossRefGoogle Scholar
  13. 13.
    J. Labetoulle, E.L. Lawler, J.K. Lenstra and A.H.G. Rinnooy Kan, Preemptive scheduling of uniform machines subject to release dates, in: W.R. Pulleyblank (ed.), Progress in Combinatorial Optimization, Academic Press, New York, 1984, 245–261.Google Scholar
  14. 14.
    L. Lovasz, On the ratio of optimal integral and fractional covers, Discrete Mathematics 13(1975), 383–390.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, Journal of the ACM 41 (1994), 960–981.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Klaus Jansen
    • 1
  • Lorant Porkolab
    • 2
  1. 1.Christian Albrechts University of KielGermany
  2. 2.Imperial CollegeLondonUK

Personalised recommendations