Precedence-Constrained Scheduling Problems Parameterized by Partial Order Width

Abstract

Negatively answering a question posed by Mnich and Wiese (Math. Program. 154(1–2):533–562), we show that $\hbox {P2}|\hbox {prec}, p_{j}{\in }\{1,2\}|C_{\max }$, 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.

R. van Bevern—Supported by project 16-31-60007 mol_a_dk of the Russian Foundation for Basic Research.

C. Komusiewicz—Supported by the DFG, project MAGZ (KO 3669/4-1).

N. Talmon—Supported by a postdoctoral fellowship from I-CORE ALGO.