Skip to main content
Log in

Preemptive Scheduling with Dedicated Processors: Applications of Fractional Graph Coloring

  • Published:
Journal of Scheduling Aims and scope Submit manuscript

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.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  • Bianco, L., and J. Blazewicz, P. Dell'Olmo, M. Drozdowski, “Scheduling preemptive multiprocessor tasks on dedicated processors,” Perform. Eval., 20, 361–371 (1994).

    Google Scholar 

  • Bianco, L., J. Blazewicz, P. Dell'Olmo, M. Drozdowski, “Scheduling multiprocessor tasks on a dynamic configuration of dedicated processors,” Ann. Oper. Res., 58, 493–517 (1995).

    Google Scholar 

  • Bianco, L., J. Blazewicz, P. Dell'Olmo, and M. Drozdowski, “Preemptive multiprocessor task scheduling with release and time windows,” Ann. Oper. Res., 70, 43–55 (1997).

    Google Scholar 

  • Blazewicz, J., P. Dell'Olmo, M. Drozdowski, and M. G. Speranza, “Scheduling multiprocessor tasks on three dedicated processors,” Inform. Process. Lett., 41, 275–280 (1992). Corrigendum: Inform. Process. Lett., 49 269–270 (1994).

    Google Scholar 

  • Blazewicz, J., M. Drozdowski, D. de Werra, and J. Weglarz, “Deadline scheduling of multiprocessor tasks,” Discrete Appl. Math., 65, 81–95 (1996).

    Google Scholar 

  • Blazewicz, J., K. H. Ecker, E. Pesch, G. Schmidt, and J. Weglarz, Scheduling Computer and Manufacturing Processes, Springer Verlag, Berlin, 1996.

    Google Scholar 

  • Drozdowski, M., “Scheduling multiprocessor tasks—an overview,” Eur. J. Oper. Res., 94, 215–230 (1996).

    Google Scholar 

  • Feige, U. and J. Kilian, “Zero knowledge and the chromatic number,” J. Comput. Syst. Sci., 57, 187–199 (1998).

    Google Scholar 

  • Grötschel, M., L. Lovasz, and A. Schrijver, “The Ellipsoid Method and its consequences in combinatorial optimization,” Combinatorica, 1, 169–197 (1981).

    Google Scholar 

  • Grötschel, M., L. Lovasz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer Verlag, Berlin, 1988.

    Google Scholar 

  • Hoogeveen, J. A., S. L. van de Velde, and B. Veltman, “Complexity of scheduling multiprocessor tasks with prespecified processor allocations,” Discrete Appl. Math., 55, 259–272 (1994).

    Google Scholar 

  • Jansen, K. and L. Porkolab, “General multiprocessor task scheduling: approximate solutions in linear time,” in Proc. 6th International Workshop on Data Structures and Algorithms (WADS '99), LNCS 1663, Springer, 1999, pp. 110–121.

  • Krämer, A., “Scheduling multiprocessor tasks on dedicated processors,” Ph.D. thesis, Fachbereich Mathematik—Informatik, Universität Osnabrück, Germany, 1995.

    Google Scholar 

  • Krawczyk, H. and M. Kubale, “An approximation algorithm for diagnostic test scheduling in multicomputer systems,” IEEE T. Comput., 34, 869–872 (1985).

    Google Scholar 

  • Kubale, M., “Preemptive versus nonpreemptive scheduling of biprocessor tasks on dedictated processors,” European J. Oper. Res., 94, 242–251 (1996).

    Google Scholar 

  • Labetoulle, J., 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, pp. 245–26.

    Google Scholar 

  • Lovasz, L., “On the ratio of optimal integral and fractional covers,” Discrete Math., 13, 383–390 (1975).

    Google Scholar 

  • Lund, C. and M. Yannakakis, “On the hardness of approximating minimization problems,” J. ACM, 41, 960–981 (1994).

    Google Scholar 

  • Papadimitriou, C. H., Computational Complexity, Addison-Wesley, Reading, MA, 1994.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Klaus Jansen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jansen, K., Porkolab, L. Preemptive Scheduling with Dedicated Processors: Applications of Fractional Graph Coloring. Journal of Scheduling 7, 35–48 (2004). https://doi.org/10.1023/B:JOSH.0000013054.30334.b9

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JOSH.0000013054.30334.b9

Navigation