Makespan minimization with OR-precedence constraints

Felix Happach ()
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Felix Happach: Technische Universität München

Journal of Scheduling, 2021, vol. 24, issue 3, No 4, 319-328

Abstract: 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.

Keywords: Scheduling; Precedence constraints; Approximation algorithm; Makespan (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10951-021-00687-6

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