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Makespan minimization with OR-precedence constraints

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  • Felix Happach

    (Technische Universität München)

Abstract

We consider a variant of the NP-hard problem of assigning jobs to machines to minimize the completion time of the last job. Usually, precedence constraints are given by a partial order on the set of jobs, and each job requires all its predecessors to be completed before it can start. In this paper, we consider a different type of precedence relation that has not been discussed as extensively and is called OR-precedence. In order for a job to start, we require that at least one of its predecessors is completed—in contrast to all its predecessors. Additionally, we assume that each job has a release date before which it must not start. We prove that a simple List Scheduling algorithm due to Graham (Bell Syst Tech J 45(9):1563–1581, 1966) has an approximation guarantee of 2 and show that obtaining an approximation factor of $$4/3 - \varepsilon $$ 4 / 3 - ε is NP-hard. Further, we present a polynomial-time algorithm that solves the problem to optimality if preemptions are allowed. The latter result is in contrast to classical precedence constraints where the preemptive variant is already NP-hard. Our algorithm generalizes previous results for unit processing time jobs subject to OR-precedence constraints, but without release dates. The running time of our algorithm is $$O(n^2)$$ O ( n 2 ) for arbitrary processing times and it can be reduced to O(n) for unit processing times, where n is the number of jobs. The performance guarantees presented here match the best-known ones for special cases where classical precedence constraints and OR-precedence constraints coincide.

Suggested Citation

  • Felix Happach, 2021. "Makespan minimization with OR-precedence constraints," Journal of Scheduling, Springer, vol. 24(3), pages 319-328, June.
    • Handle: RePEc:spr:jsched:v:24:y:2021:i:3:d:10.1007_s10951-021-00687-6
    DOI: 10.1007/s10951-021-00687-6
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    References listed on IDEAS

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    1. Robert McNaughton, 1959. "Scheduling with Deadlines and Loss Functions," Management Science, INFORMS, vol. 6(1), pages 1-12, October.
    2. T. C. Hu, 1961. "Parallel Sequencing and Assembly Line Problems," Operations Research, INFORMS, vol. 9(6), pages 841-848, December.
    3. Clyde L. Monma, 1982. "Linear-Time Algorithms for Scheduling on Parallel Processors," Operations Research, INFORMS, vol. 30(1), pages 116-124, February.
    4. Peter Brucker & M. R. Garey & D. S. Johnson, 1977. "Scheduling Equal-Length Tasks Under Treelike Precedence Constraints to Minimize Maximum Lateness," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 275-284, August.
    5. J. K. Lenstra & A. H. G. Rinnooy Kan, 1978. "Complexity of Scheduling under Precedence Constraints," Operations Research, INFORMS, vol. 26(1), pages 22-35, February.
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    Cited by:

    1. Luo, Kaiping & Shen, Guangya & Li, Liheng & Sun, Jianfei, 2023. "0-1 mathematical programming models for flexible process planning," European Journal of Operational Research, Elsevier, vol. 308(3), pages 1160-1175.

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