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Normal-form preemption sequences for an open problem in scheduling theory

Author

Listed:
  • Bo Chen

    (University of Warwick)

  • Ed Coffman

    (Columbia University)

  • Dariusz Dereniowski

    (Gdańsk University of Technology)

  • Wiesław Kubiak

    (Memorial University)

Abstract

Structural properties of optimal preemptive schedules have been studied in a number of recent papers with a primary focus on two structural parameters: the minimum number of preemptions necessary, and a tight lower bound on shifts, i.e., the sizes of intervals bounded by the times created by preemptions, job starts, or completions. These two parameters have been investigated for a large class of preemptive scheduling problems, but so far only rough bounds for these parameters have been derived for specific problems. This paper sharpens the bounds on these structural parameters for a well-known open problem in the theory of preemptive scheduling: Instances consist of in-trees of n unit-execution-time jobs with release dates, and the objective is to minimize the total completion time on two processors. This is among the current, tantalizing “threshold” problems of scheduling theory: Our literature survey reveals that any significant generalization leads to an NP-hard problem, but that any significant, but slight simplification leads to tractable problem with a polynomial-time solution. For the above problem, we show that the number of preemptions necessary for optimality need not exceed $$2n-1$$ 2 n - 1 ; that the number must be of order $${\varOmega }(\log n)$$ Ω ( log n ) for some instances; and that the minimum shift need not be less than $$2^{-2n+1}.$$ 2 - 2 n + 1 . These bounds are obtained by combinatorial analysis of optimal preemptive schedules rather than by the analysis of polytope corners for linear-program formulations of the problem, an approach to be found in earlier papers. The bounds immediately follow from a fundamental structural property called normality, by which minimal shifts of a job are exponentially decreasing functions. In particular, the first interval between a preempted job’s start and its preemption must be a multiple of 1 / 2, the second such interval must be a multiple of 1 / 4, and in general, the i-th preemption must occur at a multiple of $$2^{-i}$$ 2 - i . We expect the new structural properties to play a prominent role in finally settling a vexing, still-open question of complexity.

Suggested Citation

  • Bo Chen & Ed Coffman & Dariusz Dereniowski & Wiesław Kubiak, 2016. "Normal-form preemption sequences for an open problem in scheduling theory," Journal of Scheduling, Springer, vol. 19(6), pages 701-728, December.
  • Handle: RePEc:spr:jsched:v:19:y:2016:i:6:d:10.1007_s10951-015-0446-9
    DOI: 10.1007/s10951-015-0446-9
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    References listed on IDEAS

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    1. Peter Brucker & Johann Hurink & Sigrid Knust, 2003. "A polynomial algorithm for P | p j =1, r j , outtree | ∑ C j," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 56(3), pages 407-412, January.
    2. Yumei Huo & Joseph Leung, 2005. "Minimizing total completion time for UET tasks with release time and outtree precedence constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 62(2), pages 275-279, November.
    3. Lee A. Herrbach & Joseph Y.-T. Leung, 1990. "Preemptive Scheduling of Equal Length Jobs on Two Machines to Minimize Mean Flow Time," Operations Research, INFORMS, vol. 38(3), pages 487-494, June.
    4. Philippe Baptiste & Vadim G. Timkovsky, 2004. "Shortest path to nonpreemptive schedules of unit-time jobs on two identical parallel machines with minimum total completion time," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 60(1), pages 145-153, September.
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    Cited by:

    1. Tianyu Wang & Odile Bellenguez, 2021. "Three notes on scheduling unit-length jobs with precedence constraints to minimize the total completion time," Journal of Scheduling, Springer, vol. 24(6), pages 649-662, December.

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