eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
18:1
18:16
10.4230/LIPIcs.IPEC.2022.18
article
Hardness of Interval Scheduling on Unrelated Machines
Hermelin, Danny
1
Itzhaki, Yuval
1
Molter, Hendrik
1
Shabtay, Dvir
1
Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel
We provide new (parameterized) computational hardness results for Interval Scheduling on Unrelated Machines. It is a classical scheduling problem motivated from just-in-time or lean manufacturing, where the goal is to complete jobs exactly at their deadline. We are given n jobs and m machines. Each job has a deadline, a weight, and a processing time that may be different on each machine. The goal is find a schedule that maximizes the total weight of jobs completed exactly at their deadline. Note that this uniquely defines a processing time interval for each job on each machine.
Interval Scheduling on Unrelated Machines is closely related to coloring interval graphs and has been thoroughly studied for several decades. However, as pointed out by Mnich and van Bevern [Computers & Operations Research, 2018], the parameterized complexity for the number m of machines as a parameter remained open. We resolve this by showing that Interval Scheduling on Unrelated Machines is W[1]-hard when parameterized by the number m of machines. To this end, we prove W[1]-hardness with respect to m of the special case where we have parallel machines with eligible machine sets for jobs. This answers Open Problem 8 of Mnich and van Bevern’s list of 15 open problems in the parameterized complexity of scheduling [Computers & Operations Research, 2018].
Furthermore, we resolve the computational complexity status of the unweighted version of Interval Scheduling on Unrelated Machines by proving that it is NP-complete. This answers an open question by Sung and Vlach [Journal of Scheduling, 2005].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol249-ipec2022/LIPIcs.IPEC.2022.18/LIPIcs.IPEC.2022.18.pdf
Just-in-time scheduling
Parallel machines
Eligible machine sets
W[1]-hardness
NP-hardness