René van Bevern, Robert Bredereck, Laurent Bulteau, Christian Komusiewicz, Nimrod Talmon, and Gerhard J. Woeginger. Precedence-constrained scheduling problems parameterized by partial order width. In Proceedings of the 9th International Conference on Discrete Optimization and Operations Research (DOOR'16), Vladivostok, Russian Federation, volume 9869 of Lecture Notes in Computer Science, pages 105–120. Springer, 2016.

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 minimum-makespan 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.

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