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 \(B \ge 2\), no unit time interval \([x, x+1)\) 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 \(B\ge 3\) and we analyze the worst-case behavior of the makespan \(\tau _\mathrm{LPT}(S)\) of LPT (longest processing time first) schedules (where we rearrange the set S of jobs into non-increasing order) in relation to the makespan \(\tau _o(S)\) of optimal schedules. We show that LPT ordered jobs can be processed within a factor of \(2-2/B\) of the optimum (plus 1) and that this factor is best possible.
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References
Braun O, Chung F, Graham RL (2014a) Single processor scheduling with time restrictions. J Sched 17:399–403
Braun O, Chung F, Graham RL (2014b) Bounds on single processor scheduling with time restrictions. In: Fliedner T, Kolisch R, Naber A (eds) Proceedings of the 14th International Conference on Project Management and Scheduling, pp 48–51
Garey MR, Johnson DS (1979) Computers and Intractability: a guide to the theory of np-completeness W.H. Freeman, New York
Graham RL (1966) Bounds for certain multiprocessing anomalies. Bell Syst Tech J 45:1563–1581
Graham RL (1969) Bounds on multiprocessing timing anomalies. SIAM J Appl Math 17:416–429
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Braun, O., Chung, F. & Graham, R. Worst-case analysis of the LPT algorithm for single processor scheduling with time restrictions. OR Spectrum 38, 531–540 (2016). https://doi.org/10.1007/s00291-016-0431-5
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DOI: https://doi.org/10.1007/s00291-016-0431-5