Scheduling with two non-unit task lengths is NP-complete

Abstract

The non-preemptive job scheduling problem with release times and deadlines on a single machine is fundamental to many scheduling problems. We parameterize this problem by the set of job lengths the jobs can have. The case where all job lengths are identical is known to be solvable in polynomial time. We prove that the problem with two job lengths is NP-complete, except for the case in which the short jobs have unit job length, which was already known to be efficiently solvable. The proof uses a reduction from satisfiability to an auxiliary scheduling problem that includes a set of paired jobs that each have both an early and a late deadline, and of which at least one should be scheduled before the early deadline. This reduction is enabled by not only these pairwise dependencies between jobs, but also by dependencies introduced by specifically constructed sets of jobs which have deadlines close to each other. The auxiliary scheduling problem in its turn can be reduced to the scheduling problem with two job lengths by representing each pair of jobs with two deadlines by four different jobs.