Scheduling jobs with communication delays: Using infeasible solutions for approximation

Extended abstract
  • Rolf H. Möhring
  • Markus W. Schäffter
  • Andreas S. Schulz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1136)

Abstract

In the last few years, multi-processor scheduling with interprocessor communication delays has received increasing attention. This is due to the more realistic constraints in modeling parallel processor systems.

Most research in this vein is concerned with the makespan criterion. We contribute to this work by presenting a new and simple (2−1/m)-approximation algorithm for scheduling to minimize the makespan on identical parallel processors subject to series-parallel precedence constraints and both unit processing times and communication delays. This meets the best known performance guarantee for the same problem but without communication delays. For the same problem but with (non-trivial) release dates, arbitrary precedence constraints, arbitrary processing times and “locally small” communication delays we obtain a simple 7/3-approximation algorithm compared with the involved (7/3−4/3m)-approximation algorithm by Hanen and Munier for the case with identical release dates.

Another quite important goal in real-world scheduling is to optimize average performance. Very recently, there have been significant developments in computing nearly optimal schedules for several classic processor scheduling models to minimize the average weighted completion time. In this paper, we study for the first time scheduling with communication delays to minimize the average weighted completion time. Specifically, based on an LP relaxation we give the first constant-factor polynomial-time approximation algorithm for scheduling identical parallel processors subject to release dates and locally small communication delays. Moreover, the optimal LP value provides a lower bound on the optimum with the same worst-case performance guarantee.

The common underlying idea of our algorithms is to compute first a schedule that regards all constraints except for the processor restrictions. This schedule is then used to construct a provable good feasible schedule for a given number of processors and as a tool in the analysis of our algorithms. Complementing our approximation results, we also show that minimizing the makespan on an unrestricted number of identical parallel processors subject to series-parallel precedence constraints, unit-time jobs, and zero-one communication delays is NP-hard.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [CP95]
    P. Chrétienne and C. Picouleau, Scheduling with communication delays: a survey, Scheduling Theory and its Applications (P. Chrétienne, E. G. Coffman Jr, J. K. Lenstra, and Z. Liu, eds.), John Wiley & Sons, 1995, pp. 65–90.Google Scholar
  2. [CPS+96]
    S. Chakrabarti, C. A. Phillips, A. S. Schulz, D. B. Shmoys, C. Stein, and J. Wein, Improved scheduling algorithms for minsum criteria, 1996, To appear in Springer Lecture Notes in Computer Science, Proceedings of the 23rd ICALP Conference.Google Scholar
  3. [GK96]
    M. X. Goemans and J. Kleinberg, An improved approximation ratio for the minimum latency problem, Proceedings of the 7th ACM-SIAM Symposium on Discrete Algorithms, 1996.Google Scholar
  4. [GLS88]
    M. Grötschel, L. Lovász, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Algorithms and Combinatorics, vol. 2, Springer, Berlin, 1988.Google Scholar
  5. [Gra66]
    R. L. Graham, Bounds for certain multiprocessing anomalies, Bell System Tech. J. 45 (1966), 1563–1581.Google Scholar
  6. [HM95]
    C. Hanen and A. Munier, An approximation algorithm for scheduling dependent tasks on m processors with small communication delays, Preprint, Laboratoire Informatique Théorique et Programmation, Institut Blaise Pascal, Université Pierre et Marie Curie, 1995.Google Scholar
  7. [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, University of Technology, Berlin, Germany, 1996, submitted. Available from ftp://ftp.math.tu-berlin.de/pub/Preprints/combi/Report-516-1996.ps.Z.Google Scholar
  8. [HSW96]
    L. A. Hall, D. B. Shmoys, and J. Wein, Scheduling to minimize average completion time: Off-line and on-line algorithms, Proceedings of the 7th ACM-SIAM Symposium on Discrete Algorithms, 1996, pp. 142–151.Google Scholar
  9. [HVL94]
    J. A. Hoogeveen, B. Veltman, and J. K. Lenstra, Three, four, five, six, or the complexity of scheduling with communication delays, Operations Research Letters 16 (1994), 129–137.Google Scholar
  10. [Law93]
    E. L. Lawler, Scheduling trees on multiprocessors with unit communication delays, Presented at the First Workshop on Models and Algorithms for Planning and Scheduling Problems, unpublished manuscript, June 1993.Google Scholar
  11. [LVV96]
    J. K. Lenstra, M. Veldhorst, and B. Veltman, The complexity of scheduling trees with communication delays, Journal of Algorithms 20 (1996), 157–173.Google Scholar
  12. [MK93]
    A. Munier and J.-C. König, A heuristic for a scheduling problem with communication delays, Preprint 871, Laboratoire de Recherche en informatique, Université de Paris, France, 1993, to appear in Operations Research, 1996.Google Scholar
  13. [Möh89]
    R. H. Möhring, Computationally tractable classes of ordered sets, Algorithms and Order (I. Rival, ed.), Nato Advanced Study Institutes Series, D. Reidel Publishing Company, Dordrecht, 1989, pp. 105–193.Google Scholar
  14. [MS96]
    R. H. Möhring and M. W. Schäffter, A simple approximation algorithm for scheduling forests with unit processing times and zero-one communication delays, Preprint No. 506/1996, University of Technology, Berlin, 1996, ftp://ftp.math.tuberlin.de/pub/Preprints/combi/Report-506-1995.ps.Z.Google Scholar
  15. [MSS96]
    R. H. Möhring, M. W. Schäffter, and A. S. Schulz, Scheduling jobs with communication delays: Using infeasible solutions for approximation, Preprint 517/1996, Department of Mathematics, University of Technology, Berlin, Germany, 1996.Google Scholar
  16. [Pic95]
    C. Picouleau, New complexity results on scheduling with small communication delays, Discrete Applied Mathematics 60 (1995), 331–342.Google Scholar
  17. [PSW95]
    C. Phillips, C. Stein, and J. Wein, Scheduling jobs that arrive over time, Proceedings of the Fourth Workshop on Algorithms and Data Structures (Berlin), Lecture Notes in Computer Science, no. 955, Springer, Berlin, 1995, pp. 86–97.Google Scholar
  18. [RS87]
    V. J. Rayward-Smith, UET scheduling with unit interprocessor communication delays, Discrete Applied Math. 18 (1987), 55–71.Google Scholar
  19. [Sch95]
    A. S. Schulz, Polytopes and scheduling, Ph.D. thesis, University of Technology, Berlin, Germany, 1995.Google Scholar
  20. [Sch96]
    A. S. Schulz, Scheduling to minimize total weighted completion time: Performance guarantees of LP-based heuristics and lower bounds, Integer Programming and Combinatorial Optimization (Berlin) (W. H. Cunningham, S. T. McCormick, and M. Queyranne, eds.), Lecture Notes in Computer Science, no. 1084, Springer, Berlin, 1996, Proceedings of the 5th International IPCO Conference, pp. 301–315.Google Scholar
  21. [Ver95]
    J. Verriet, Scheduling UET, UCT dags with release dates and deadlines, Preprint No. UU-CS-1995-31, Utrecht University, Department of Computer Science, 1995.Google Scholar
  22. [VLL90]
    B. Veltman, B. Lageweg, and J. K. Lenstra, Multiprocessor scheduling with commmunication delays, Parallel Computing 16 (1990), 173–182.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Rolf H. Möhring
    • 1
  • Markus W. Schäffter
    • 1
  • Andreas S. Schulz
    • 1
  1. 1.Fachbereich MathematikTechnische Universität BerlinBerlinGermany

Personalised recommendations