Abstract
In this paper, for the problem of minimizing the makespan on two unrelated parallel machines we compare the quality of preemptive and non-preemptive schedules. It is known that there exists an optimal preemptive schedule with at most two preemptions. We show that the power of preemption, i.e., the ratio of the makespan computed for the best non-preemptive schedule to the makespan of the optimal preemptive schedule is at most 3/2. We also show that the ratio of the makespan computed for the best schedule with at most one preemption to the makespan of the optimal preemptive schedule is at most 9/8. For both models, we present polynomial-time algorithms that find schedules of the required quality. The established bounds match those previously known for a less general problem with two uniform machines. We have found one point of difference between the uniform and unrelated machines: if an optimal preemptive schedule contains exactly one preemption then the ratio of the makespan computed for the best non-preemptive schedule to the makespan of the optimal preemptive schedule is at most 4/3 if the two machines are uniform and remains 3/2 if the machines are unrelated.
Similar content being viewed by others
References
Braun, O., & Schmidt, G. (2003). Parallel processor scheduling with limited number of preemptions. SIAM Journal on Computing, 32, 671–680.
Chen, B. (2004). Parallel machine scheduling for early completion. In J.Y.-T. Leung (Ed.), Handbook of scheduling: Algorithms, models and performance analysis (pp. 9-175–9-184). London: Chapman & Hall/CRC.
Correa, J. R., Skutella, M., & Verschae, J. (2012). The power of preemption on unrelated machines and applications to scheduling orders. Mathematics of Operations Research, 37, 379–398.
Epstein, L., & Levin, A. (2016). The benefit of preemption for single machine scheduling so as to minimize total weighted completion time. Operations Research Letters, 44, 772–774.
Epstein, L., Levin, A., Soper, A. J., & Strusevich, V. A. (2017). Power of preemption for minimizing total completion time on uniform parallel machines. SIAM Journal on Discrete Mathematics, 31, 101–123.
Gonzalez, T. F., Lawler, E. L., & Sahni, S. (1990). Optimal preemptive scheduling of two unrelated processors. ORSA Journal on Computing, 2(3), 219–224.
Gonzalez, T. F., & Sahni, S. (1978). Preemptive scheduling of uniform processor systems. Journal of the Association for Computing Machinery, 25, 92–101.
Jiang, Y., Weng, Z., & Hu, J. (2014). Algorithms with limited number of preemptions for scheduling on parallel machines. Journal of Combinatorial Optimization, 27, 711–723.
Lawler, E. L., & Labetoulle, J. (1978). On preemptive scheduling of unrelated parallel processors by linear programming. Journal of the Association for Computing Machinery, 25(4), 612–619.
Lee, C.-Y., & Strusevich, V. A. (2005). Two-machine shop scheduling with an uncapacitated interstage transporter. IIE Transactions, 37, 725–736.
Lenstra, J. K., Shmoys, D. B., & Tardos, E. (1990). Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming, 46, 259–271.
Lin, J.-H., & Vitter, J. S. (1992). \(\varepsilon \)-approximations with minimum packing constraint violation. In Proceedings of the 24th annual ACM symposium on theory of computing (STOC) (pp. 771–782). New York: ACM.
McNaughton, R. (1959). Scheduling with deadlines and loss functions. Management Science, 6, 1–12.
Potts, C. N. (1985). Analysis of a linear programming heuristic for scheduling unrelated parallel machines. Discrete Applied Mathematics, 10, 155–164.
Rustogi, K., & Strusevich, V. A. (2013). Parallel machine scheduling: Impact of adding extra machines. Operations Research, 61, 1243–1257.
Shchepin, E., & Vakhania, N. (2005). An optimal rounding gives a better approximation for scheduling unrelated machines. Operations Research Letters, 33, 127–133.
Shchepin, E., & Vakhania, N. (2008). On the geometry, preemptions and complexity of multiprocessor and shop scheduling. Annals of Operations Research, 159, 183–213.
Shmoys, D. B., & Tardos, E. (1993). An approximation algorithm for the generalized assignment problem. Mathematical Programming, 62, 461–474.
Sitters, R. A. (2017). Approximability of average completion time scheduling on unrelated parallel machines. Mathematical Programming, 161, 135–158.
Soper, A. J., & Strusevich, V. A. (2014a). Single parameter analysis of power of preemption on two and three uniform machines. Discrete Optimization, 12, 26–46.
Soper, A. J., & Strusevich, V. A. (2014b). Power of preemption on uniform parallel machines. In 17th International workshop on approximation algorithms for combinatorial optimization problems (APPROX’14)/18th international workshop on randomization and computation (RANDOM’14). Leibniz International Proceedings in Informatics (LIPIcs) (vol. 28, pp. 392–402).
Soper, A. J., & Strusevich, V. A. (2019). Schedules with a single preemption on uniform parallel machines. Discrete Applied Mathematics, 261, 332–343.
Soper, A. J., & Strusevich, V. A. (2021). Parametric analysis of the quality of single preemption schedules on three uniform parallel machines. Annals of Operations Research, 298, 469–495.
Woeginger, G. J. (2000). A comment on scheduling on uniform machines under chain-like precedence constraints. Operations Research Letters, 26, 107–109.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Soper, A.J., Strusevich, V.A. Preemptive and non-preemptive scheduling on two unrelated parallel machines. J Sched 25, 659–674 (2022). https://doi.org/10.1007/s10951-022-00753-7
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10951-022-00753-7