Negatively answering a question posed by Mnich and Wiese (Math. Program. 154(1–2):533–562), we show that P2|prec, pj∈{1, 2}|Cmax, the problem of finding a non-preemptive minimummakespan schedule for precedence-constrained jobs of lengths 1 and 2 on two parallel identical machines, is W[2]-hard parameterized by the width of the partial order giving the precedence constraints. To this end, we show that Shuffle Product, the problem of deciding whether a given word can be obtained by interleaving the letters of k other given words, is W[2]-hard parameterized by k, thus additionally answering a question posed by Rizzi and Vialette (CSR 2013). Finally, refining a geometric algorithm due to Servakh (Diskretn. Anal. Issled. Oper. 7(1):75–82), we show that the more general Resource-Constrained Project Scheduling problem is fixed-parameter tractable parameterized by the partial order width combined with the maximum allowed difference between the earliest possible and factual starting time of a job.
CITATION STYLE
van Bevern, R., Bredereck, R., Bulteau, L., Komusiewicz, C., Talmon, N., & Woeginger, G. J. (2016). Precedence-constrained scheduling problems parameterized by partial order width. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9869 LNCS, pp. 105–120). Springer Verlag. https://doi.org/10.1007/978-3-319-44914-2_9
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