Abstract
In this paper, we consider the parameterized complexity of the following scheduling problem. We must schedule a number of jobs on m machines, where each job has unit length, and the graph of precedence constraints consists of a set of chains. Each precedence constraint is labelled with an integer that denotes the exact (or minimum) delay between the jobs. We study different cases; delays can be given in unary and in binary, and the case that we have a single machine is discussed separately. We consider the complexity of this problem parameterized by the number of chains, and by the thickness of the instance, which is the maximum number of chains whose intervals between release date and deadline overlap.
We show that this scheduling problem with exact delays in unary is W[t]hard for all t, when parameterized by the thickness, even when we have a single machine (m = 1). When parameterized by the number of chains, this problem is W[1]complete when we have a single or a constant number of machines, and W[2]complete when the number of machines is a variable. The problem with minimum delays, given in unary, parameterized by the number of chains (and as a simple corollary, also when parameterized by the thickness) is W[1]hard for a single or a constant number of machines, and W[2]hard when the number of machines is variable.
With a dynamic programming algorithm, one can show membership in XP for exact and minimum delays in unary, for any number of machines, when parameterized by thickness or number of chains. For a single machine, with exact delays in binary, parameterized by the number of chains, membership in XP can be shown with branching and solving a system of difference constraints. For all other cases for delays in binary, membership in XP is open.
BibTeX  Entry
@InProceedings{bodlaender_et_al:LIPIcs:2020:13307,
author = {Hans L. Bodlaender and Marieke van der Wegen},
title = {{Parameterized Complexity of Scheduling Chains of Jobs with Delays}},
booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
pages = {4:14:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771726},
ISSN = {18688969},
year = {2020},
volume = {180},
editor = {Yixin Cao and Marcin Pilipczuk},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/13307},
URN = {urn:nbn:de:0030drops133075},
doi = {10.4230/LIPIcs.IPEC.2020.4},
annote = {Keywords: Scheduling, parameterized complexity}
}
Keywords: 

Scheduling, parameterized complexity 
Collection: 

15th International Symposium on Parameterized and Exact Computation (IPEC 2020) 
Issue Date: 

2020 
Date of publication: 

04.12.2020 