Abstract
In this paper we study multiprocessor and open shop scheduling problems from several points of view. We explore a tight dependence of the polynomial solvability/intractability on the number of allowed preemptions. For an exhaustive interrelation, we address the geometry of problems by means of a novel graphical representation. We use the so-called preemption and machine-dependency graphs for preemptive multiprocessor and shop scheduling problems, respectively. In a natural manner, we call a scheduling problem acyclic if the corresponding graph is acyclic. There is a substantial interrelation between the structure of these graphs and the complexity of the problems. Acyclic scheduling problems are quite restrictive; at the same time, many of them still remain NP-hard. We believe that an exhaustive study of acyclic scheduling problems can lead to a better understanding and give a better insight of general scheduling problems.
We show that not only acyclic but also a special non-acyclic version of periodic job-shop scheduling can be solved in polynomial (linear) time. In that version, the corresponding machine dependency graph is allowed to have a special type of the so-called parti-colored cycles. We show that trivial extensions of this problem become NP-hard. Then we suggest a linear-time algorithm for the acyclic open-shop problem in which at most m−2 preemptions are allowed, where m is the number of machines. This result is also tight, as we show that if we allow one less preemption, then this strongly restricted version of the classical open-shop scheduling problem becomes NP-hard. In general, we show that very simple acyclic shop scheduling problems are NP-hard. As an example, any flow-shop problem with a single job with three operations and the rest of the jobs with a single non-zero length operation is NP-hard. We suggest linear-time approximation algorithm with the worst-case performance of \(\|\mathcal{M}\|+2\|\mathcal{J}\|\) (\(\|\mathcal{M}\|+\|\mathcal{J}\|\) , respectively) for acyclic job-shop (open-shop, respectively), where \(\|\mathcal{J}\|\) (‖ℳ‖, respectively) is the maximal job length (machine load, respectively). We show that no algorithm for scheduling acyclic job-shop can guarantee a better worst-case performance than \(\|\mathcal{M}\|+\|\mathcal{J}\|\) . We consider two special cases of the acyclic job-shop with the so-called short jobs and short operations (restricting the maximal job and operation length) and solve them optimally in linear time. We show that scheduling m identical processors with at most m−2 preemptions is NP-hard, whereas a venerable early linear-time algorithm by McNaughton yields m−1 preemptions. Another multiprocessor scheduling problem we consider is that of scheduling m unrelated processors with an additional restriction that the processing time of any job on any machine is no more than the optimal schedule makespan C *max . We show that the (2m−3)-preemptive version of this problem is polynomially solvable, whereas the (2m−4)-preemptive version becomes NP-hard. For general unrelated processors, we guarantee near-optimal (2m−3)-preemptive schedules. The makespan of such a schedule is no more than either the corresponding non-preemptive schedule makespan or max {C *max ,p max }, where C *max is the optimal (preemptive) schedule makespan and p max is the maximal job processing time.
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E.V. Shchepin was partially supported by the program “Algebraical and combinatorial methods of mathematical cybernetics” of the Russian Academy of Sciences.
N. Vakhania was partially supported by CONACyT grant No. 48433.
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Shchepin, E.V., Vakhania, N. On the geometry, preemptions and complexity of multiprocessor and shop scheduling. Ann Oper Res 159, 183–213 (2008). https://doi.org/10.1007/s10479-007-0266-1
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DOI: https://doi.org/10.1007/s10479-007-0266-1