Random-based scheduling new approximations and LP lower bounds

Extended abstract
  • Andreas S. Schulz
  • Martin Skutella
Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1269)

Abstract

Three characteristics encountered frequently in real-world machine scheduling are jobs released over time, precedence constraints between jobs, and average performance optimization. The general constrained one-machine scheduling problem to minimize the average weighted completion time not only captures these features, but also is an important building block for more complex problems involving multiple machines.

In this context, the conversion of preemptive to nonpreemptive schedules has been established as a strong and useful tool for the design of approximation algorithms.

The preemptive problem is already NP-hard, but one can generate good preemptive schedules from LP relaxations in time-indexed variables. However, a straightforward combination of these two components does not directly lead to improved approximations. By showing schedules in slow motion, we introduce a new point of view on the generation of preemptive schedules from LP-solutions which also enables us to give a better analysis.

Specifically, this leads to a randomized approximation algorithm for the general constrained one-machine scheduling problem with expected performance guarantee e. This improves upon the best previously known worst-case bound of 3. In the process, we also give randomized algorithms for related problems involving precedences that asymptotically match the best previously known performance guarantees.

In addition, by exploiting a different technique, we give a simple 3/2-approximation algorithm for unrelated parallel machine scheduling to minimize the average weighted completion time. It relies on random machine assignments where these random assignments are again guided by an optimum solution to an LP relaxation. For the special case of identical parallel machines, this algorithm is as simple as the one of Kawaguchi and Kyan [KK86], but allows for a remarkably simpler analysis. Interestingly, its derandomized version actually is the algorithm of Kawaguchi and Kyan.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [CMNS97]
    C. Chekuri, R. Motwani, B. Natarajan, and C. Stein. Approximation techniques for average completion time scheduling. In Proceedings of the 8th ACM—SIAM Symposium on Discrete Algorithms, pages 609–618, 1997.Google Scholar
  2. [DW90]
    M. E. Dyer and L. A. Wolsey. Formulating the single machine sequencing problem with release dates as a mixed integer program. Discrete Applied Mathematics, 26:255–270, 1990.Google Scholar
  3. [GLLRK79]
    R. L. Graham, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan. Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics, 5:287–326, 1979.Google Scholar
  4. [Goe96]
    M. X. Goemans, June 1996. Personal communication.Google Scholar
  5. [Goe97]
    M. X. Goemans. Improved approximation algorithms for scheduling with release dates. In Proceedings of the 8th ACM—SIAM Symposium on Discrete Algorithms, pages 591–598, 1997.Google Scholar
  6. [HSSW96]
    L. A. Hall, A. S. Schulz, D. B. Shmoys, and J. Wein. Scheduling to minimize average completion time: Off—line and on—line approximation algorithms. Preprint 516/1996, Department of Mathematics, Technical University of Berlin, Berlin, Germany, 1996. To appear in Mathematics of Operations Research.Google Scholar
  7. [HSW96]
    L. A. Hall, D. B. Shmoys, and J. Wein. Scheduling to minimize average completion time: Off—line and on—line algorithms. In Proceedings of the 7th ACM—SIAM Symposium on Discrete Algorithms, pages 142–151, 1996.Google Scholar
  8. [KK86]
    T. Kawaguchi and S. Kyan. Worst case bound of an LRF schedule for the mean weighted flow—time problem. SIAM Journal on Computing, 15:1119–1129, 1986.Google Scholar
  9. [Law78]
    E. L. Lawler. Sequencing jobs to minimize total weighted completion time subject to precedence constraints. Annals of Discrete Mathematics, 2:75–90, 1978.Google Scholar
  10. [PSW95]
    C. Phillips, C. Stein, and J. Wein. Scheduling jobs that arrive over time. In Proceedings of the Fourth Workshop on Algorithms and Data Structures, number 955 in Lecture Notes in Computer Science, pages 86–97. Springer, Berlin, 1995.Google Scholar
  11. [Sch96]
    A. S. Schulz. Scheduling to minimize total weighted completion time: Performance guarantees of LP—based heuristics and lower bounds. In W. H. Cunningham, S. T. McCormick, and M. Queyranne, editors, Integer Programming and Combinatorial Optimization, number 1084 in Lecture Notes in Computer Science, pages 301–315. Springer, Berlin, 1996. Proceedings of the 5th International IPCO Conference.Google Scholar
  12. [Smi56]
    W. E. Smith. Various optimizers for single—stage production. Naval Research and Logistics Quarterly, 3:59–66, 1956.Google Scholar
  13. [SS97]
    A. S. Schulz and M. Skutella. Scheduling—LPs bear probabilities: Randomized approximations formin—sum criteria. Preprint 533/1996, Department of Mathematics, Technical University of Berlin, Berlin, Germany, November 1996; revised March 1997. To appear in Springer Lecture Notes in Computer Science, Proceedings of the 5th Annual European Symposium on Algorithms (ESA'97).Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Andreas S. Schulz
    • 1
  • Martin Skutella
    • 1
  1. 1.Fachbereich MathematikTechnische Universität BerlinBerlinGermany

Personalised recommendations