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.
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REFERENCES
Bianco, L., and J. Blazewicz, P. Dell'Olmo, M. Drozdowski, “Scheduling preemptive multiprocessor tasks on dedicated processors,” Perform. Eval., 20, 361–371 (1994).
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).
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).
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).
Blazewicz, J., M. Drozdowski, D. de Werra, and J. Weglarz, “Deadline scheduling of multiprocessor tasks,” Discrete Appl. Math., 65, 81–95 (1996).
Blazewicz, J., K. H. Ecker, E. Pesch, G. Schmidt, and J. Weglarz, Scheduling Computer and Manufacturing Processes, Springer Verlag, Berlin, 1996.
Drozdowski, M., “Scheduling multiprocessor tasks—an overview,” Eur. J. Oper. Res., 94, 215–230 (1996).
Feige, U. and J. Kilian, “Zero knowledge and the chromatic number,” J. Comput. Syst. Sci., 57, 187–199 (1998).
Grötschel, M., L. Lovasz, and A. Schrijver, “The Ellipsoid Method and its consequences in combinatorial optimization,” Combinatorica, 1, 169–197 (1981).
Grötschel, M., L. Lovasz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer Verlag, Berlin, 1988.
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).
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.
Krawczyk, H. and M. Kubale, “An approximation algorithm for diagnostic test scheduling in multicomputer systems,” IEEE T. Comput., 34, 869–872 (1985).
Kubale, M., “Preemptive versus nonpreemptive scheduling of biprocessor tasks on dedictated processors,” European J. Oper. Res., 94, 242–251 (1996).
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.
Lovasz, L., “On the ratio of optimal integral and fractional covers,” Discrete Math., 13, 383–390 (1975).
Lund, C. and M. Yannakakis, “On the hardness of approximating minimization problems,” J. ACM, 41, 960–981 (1994).
Papadimitriou, C. H., Computational Complexity, Addison-Wesley, Reading, MA, 1994.
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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
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DOI: https://doi.org/10.1023/B:JOSH.0000013054.30334.b9