Interval Selection with Machine-Dependent Intervals

Abstract

We study an offline interval scheduling problem where every job has exactly one associated interval on every machine. To schedule a set of jobs, exactly one of the intervals associated with each job must be selected, and the intervals selected on the same machine must not intersect. We show that deciding whether all jobs can be scheduled is NP-complete already in various simple cases. In particular, by showing the NP-completeness for the case when all the intervals associated with the same job end at the same point in time (also known as just-in-time jobs), we solve an open problem posed by Sung and Vlach (J. Sched., 2005). Furthermore, we show NP-completeness for the variant with unit-length intervals where all intervals associated with the same job have a common point, and for the variant with unitlength intervals and three machines. We also study the related problem of maximizing the number of scheduled jobs. We prove that the problem is NP-hard even for two machines and unit-length intervals. We present a 2/3-approximation algorithm for two machines (and intervals of arbitrary lengths).

No-Wait Scheduling for Locks

“…We state that the unidirectional case of this variant is NP complete (Section 5). This result strengthens both the result given in Krumke et al (2011) and a result in Böhmová et al (2013). Finally, not all instances will admit a feasible solution where no ship incurs any waiting time.…”

“…□ Remark 4. As mentioned in Section 1, when viewing a chamber as a machine and an arrival as a job represented by m intervals (one for each machine; each starting at the same moment t i ), the reduction referred to above shows that the problem considered by Böhmová et al (2013) (called interval selection with cores) remains NP complete even when all intervals that correspond to the same machine have the same length.…”

“…The unidirectional case with distinct arrival times is also related to a problem studied in Böhmová et al (2013). In their setting, a job corresponds to a set of intervals: one for each machine.…”

No-Wait Scheduling for Locks

“…413-428, © 2019 is NP complete (Section 5). This result strengthens both the result given in Krumke et al (2011) and a result in Böhmová et al (2013). Finally, not all instances will admit a feasible solution where no ship incurs any waiting time.…”