Journal of Scheduling

, Volume 14, Issue 2, pp 141–156 | Cite as

On robust online scheduling algorithms

Article

Abstract

While standard parallel machine scheduling is concerned with good assignments of jobs to machines, we aim to understand how the quality of an assignment is affected if the jobs’ processing times are perturbed and therefore turn out to be longer (or shorter) than declared. We focus on online scheduling with perturbations occurring at any time, such as in railway systems when trains are late. For a variety of conditions on the severity of perturbations, we present bounds on the worst case ratio of two makespans. For the first makespan, we let the online algorithm assign jobs to machines, based on the non-perturbed processing times. We compute the makespan by replacing each job’s processing time with its perturbed version while still sticking to the computed assignment. The second is an optimal offline solution for the perturbed processing times. The deviation of this ratio from the competitive ratio of the online algorithm tells us about the “price of perturbations”. We analyze this setting for Graham’s algorithm, and among other bounds show a competitive ratio of 2 for perturbations decreasing the processing time of a job arbitrarily, and a competitive ratio of less than 2.5 for perturbations doubling the processing time of a job. We complement these results by providing lower bounds for any online algorithm in this setting. Finally, we propose a risk-aware online algorithm tailored for the possible bounded increase of the processing time of one job, and we show that this algorithm can be worse than Graham’s algorithm in some cases.

Keywords

Robustness Online Scheduling Graham’s Algorithm Perturbation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network flows: theory, algorithms and applications. New York: Prentice Hall. Google Scholar
  2. Albers, S. (2002). On randomized online scheduling. In STOC ’02: Proceedings of the 34th annual ACM symposium on theory of computing (pp. 134–143). New York, NY, USA, 2002. New York: ACM Press. ISBN 1-58113-495-9. CrossRefGoogle Scholar
  3. Becchetti, L., Leonardi, S., Marchetti-Spaccamela, A., Schafer, G., & Vredeveld, T. (2006). Average-case and smoothed competitive analysis of the multilevel feedback algorithm. Mathematics of Operations Research, 31(1), 85–108. CrossRefGoogle Scholar
  4. Ben-Tal, A., & Nemirovski, A. (2002). Robust optimization—methodology and applications. Mathematical Programming, 92(3), 453–480. CrossRefGoogle Scholar
  5. Chvátal, V. (1983). Linear programming. New York: Freeman. Google Scholar
  6. Fleischer, R., & Wahl, M. (2000). Online scheduling revisited. Journal of Scheduling, 3(6), 343–353. Special issue on approximation algorithms for scheduling algorithms (part 2). CrossRefGoogle Scholar
  7. Graham, R. L. (1966). Bounds for certain multiprocessor anomalies. Bell System Technical Journal, 45(9), 1563–1581. Google Scholar
  8. Graham, R. L. (1969). Bounds on multiprocessing timing anomalies. SIAM Journal on Applied Mathematics, 17(2), 416–429. CrossRefGoogle Scholar
  9. Hall, N. G., & Posner, M. E. (2004). Sensitivity analysis for scheduling problems. Journal of Scheduling, 7, 49–83. CrossRefGoogle Scholar
  10. Kasperski, A., & Zieliński, P. (2006). An approximation algorithm for interval data minmax regret combinatorial optimization problems. Information Processing Letters, 97(5), 177–180. ISSN 0020-0190. CrossRefGoogle Scholar
  11. Kolen, A. W. J., Rinnooy Kan, A. H. G., van Hoesel, C. S. M., & Wagelmans, A. P. M. (1994). Sensitivity analysis of list scheduling heuristics. Discrete Applied Mathematics, 55, 145–162. CrossRefGoogle Scholar
  12. Kouvelis, P., & Yu, G. (1997). Robust discrete optimization and its applications. Dordrecht: Kluwer Academic. Google Scholar
  13. Mauroy, G., Wardi, Y., & Proth, J. M. (1997). Sensitivity analysis of machine schedules with multi-priority job classes. In Proceedings of the 36th IEEE conference on decision and control (Vol. 5067, pp. 686–691). Google Scholar
  14. Megow, N., Uetz, M., & Vredeveld, T. (2006). Models and algorithms for stochastic online scheduling. Mathematics of Operations Research, 31(3), 513–525. CrossRefGoogle Scholar
  15. Möhring, R. H., Schulz, A. S., & Uetz, M. (1999). Approximation in stochastic scheduling: the power of LP-based priority policies. Journal of the ACM, 46(6), 924–942. CrossRefGoogle Scholar
  16. Montemanni, R., & Gambardella, L. M. (2004). An exact algorithm for the robust shortest path problem with interval data. Computers & Operations Research, 31, 1667–1680. CrossRefGoogle Scholar
  17. Moukrim, A., Sanlaville, E., & Guinan, F. (2003). Parallel machine scheduling with uncertain communication delays. RAIRO Operations Research, 37, 1–16. CrossRefGoogle Scholar
  18. Penz, B., Rapine, C., & Trystram, D. (2001). Sensitivity analysis of scheduling algorithms. European Journal of Operational Research, 134, 606–615. CrossRefGoogle Scholar
  19. Pinedo, M. (2002). Scheduling: Theory, algorithms, and systems. New York: Prentice Hall. Google Scholar
  20. Sanlaville, E. (2005). Sensitivity bounds for machine scheduling with uncertain communication delays. Journal of Scheduling, 8(5), 461–473. CrossRefGoogle Scholar
  21. Schäfer, G. (2004). Worst case instances are fragile. Average case and smoothed competitive analysis of algorithms. Ph.D. thesis, Universität des Saarlandes, April 2004. Google Scholar
  22. Scharbrodt, M., Schickinger, T., & Steger, A. (2006). A new average case analysis for completion time scheduling. Journal of the ACM, 53(1), 121–146. CrossRefGoogle Scholar
  23. Sgall, J. (1998). On-line scheduling. In A. Fiat & G. J. Woeginger (Eds.), Lecture notes in computer science : Vol. 1442. Online algorithms: The state of the art (pp. 196–231). Berlin: Springer. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.ETH ZurichZurichSwitzerland

Personalised recommendations