Precedence-constrained scheduling problems parameterized by partial order width

TL;DR

This paper proves the W[2]-hardness of a specific precedence-constrained scheduling problem with jobs of lengths 1 and 2 on two machines, and explores parameterized complexity results for related problems.

Contribution

It establishes the W[2]-hardness of P2|prec,p_j∈{1,2}|Cmax parameterized by partial order width, and refines the complexity analysis of resource-constrained scheduling.

Findings

Proves W[2]-hardness of scheduling with precedence constraints and job lengths 1 and 2 on two machines.

Shows W[2]-hardness of Shuffle Product problem parameterized by the number of words.

Demonstrates fixed-parameter tractability of resource-constrained scheduling with combined parameters.

Abstract

Negatively answering a question posed by Mnich and Wiese (Math. Program. 154(1-2):533-562), we show that P2|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.