Abstract
In this article, we investigate the parameterized complexity of coupled-task scheduling in the presence of compatibility constraints given by a compatibility graph. In this model, each task contains two sub-tasks delayed by an idle time. Moreover, a sub-task can be performed during the idle time of another task if the two tasks are compatible. We consider a parameterized version of the scheduling problem: is there a schedule in which at least k coupled-tasks have a completion time before a fixed due date? It is known that this problem is \(\mathsf { NP}\)-complete. We prove that it is fixed-parameter tractable (\(\mathsf {FPT}\)) parameterized by k the standard parameter if the total duration of each task is bounded by a constant, whereas the problem becomes \({\mathsf {W}}[1]\)-hard otherwise. We also show that in the former case, the problem does not admit a polynomial kernel under some standard complexity assumptions. Moreover, we obtain an \(\mathsf {FPT}\) algorithm when the problem is parameterized by the size of a vertex cover of the compatibility graph.
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Notes
In all the figures, the two sub-tasks of a single task are colored with the same pattern.
where \(\alpha \) designates the environment processors, \(\beta \) the characteristics of the jobs and \(\gamma \) the criteria (Graham et al. 1979).
We give here a simplified definition of a cross-composition, which it is sufficient for our purpose.
\(n(G_c)\) (resp. \(m(G_c)\)) represents the number of vertices (resp. edges) of \(G_c\).
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The authors would like to thank both anonymous referees of the journal for their helpful corrections and suggestions which improved the readability of this article.
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Bessy, S., Giroudeau, R. Parameterized complexity of a coupled-task scheduling problem. J Sched 22, 305–313 (2019). https://doi.org/10.1007/s10951-018-0581-1
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DOI: https://doi.org/10.1007/s10951-018-0581-1