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Worst-case analysis of the LPT algorithm for single processor scheduling with time restrictions

Author

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  • Oliver Braun

    (Trier University of Applied Sciences, Environmental Campus Birkenfeld)

  • Fan Chung

    (University of California)

  • Ron Graham

    (University of California)

Abstract

We consider the following scheduling problem. We are given a set S of jobs which are to be scheduled sequentially on a single processor. Each job has an associated processing time which is required for its processing. Given a particular permutation of the jobs in S, the jobs are processed in that order with each job started as soon as possible, subject only to the following constraint: For a fixed integer $$B \ge 2$$ B ≥ 2 , no unit time interval $$[x, x+1)$$ [ x , x + 1 ) is allowed to intersect more than B jobs for any real x. There are several real world situations for which this restriction is natural. For example, suppose in addition to the jobs being executed sequentially on a single main processor, each job also requires the use of one of B identical subprocessors during its execution. Each time a job is completed, the subprocessor it was using requires one unit of time to reset itself. In this way, it is never possible for more than B jobs to be worked on during any unit interval. In Braun et al. (J Sched 17: 399–403, 2014a) it is shown that this problem is NP-hard when the value B is variable and a classical worst-case analysis of List Scheduling for this situation has been carried out. We prove a tighter bound for List Scheduling for $$B\ge 3$$ B ≥ 3 and we analyze the worst-case behavior of the makespan $$\tau _\mathrm{LPT}(S)$$ τ LPT ( S ) of LPT (longest processing time first) schedules (where we rearrange the set S of jobs into non-increasing order) in relation to the makespan $$\tau _o(S)$$ τ o ( S ) of optimal schedules. We show that LPT ordered jobs can be processed within a factor of $$2-2/B$$ 2 - 2 / B of the optimum (plus 1) and that this factor is best possible.

Suggested Citation

  • Oliver Braun & Fan Chung & Ron Graham, 2016. "Worst-case analysis of the LPT algorithm for single processor scheduling with time restrictions," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 38(2), pages 531-540, March.
  • Handle: RePEc:spr:orspec:v:38:y:2016:i:2:d:10.1007_s00291-016-0431-5
    DOI: 10.1007/s00291-016-0431-5
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    Cited by:

    1. Rachid Benmansour & Oliver Braun, 2023. "On the minimum number of resources for a perfect schedule," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 31(1), pages 191-204, March.
    2. Rainer Kolisch & Erik Demeulemeester & Rubén Ruiz Garcia & Vincent T’Kindt & Jan Węglarz, 2016. "Editorial “Project Management and Scheduling”," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 38(2), pages 279-281, March.
    3. Rachid Benmansour & Oliver Braun & Saïd Hanafi, 2019. "The single-processor scheduling problem with time restrictions: complexity and related problems," Journal of Scheduling, Springer, vol. 22(4), pages 465-471, August.

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