Minimizing total tardiness on parallel machines with preemptions

The basic scheduling problem we are dealing with in this paper is the following one. A set of jobs has to be scheduled on a set of parallel uniform machines. Each machine can handle at most one job at a time. Each job becomes available for processing at its release date. All jobs have the same execution requirement and arbitrary due dates. Each machine has a known speed. The processing of any job may be interrupted arbitrarily often and resumed later on any machine. The goal is to find a schedule that minimizes the sum of tardiness, i.e., we consider problem Q∣rj,pj=p, pmtn∣∑Tj whose complexity status was open. Recently, Tian et al. (J. Sched. 9:343–364, 2006) proposed a polynomial algorithm for problem 1∣rj,pj=p, pmtn∣∑Tj. We show that both the problem P∣ pmtn∣∑Tj of minimizing total tardiness on a set of parallel machines with allowed preemptions and the problem P∣rj,pj=p, pmtn∣∑Tj of minimizing total tardiness on a set of parallel machines with release dates, equal processing times and allowed preemptions are NP-hard. Moreover, we give a polynomial algorithm for the case of uniform machines without release dates, i.e., for problem Q∣pj=p, pmtn∣∑Tj.