Preemptive scheduling with job-dependent setup times

We investigate the problem of preemptively scheduling n jobs on m parallel machines. Whenever a machine switches from the processing of one job Jk to the processing of some other job Jj, a setup time sj is necessary; this setup time sj only depends on the job Jj which is next to be processed. The objective is to find a schedule that minimizes the makespan. This problem obviously is NP-complete. In this paper we discuss polynomial time approximation algorithms for this problem. For the general case, we give an approximation algorithm whose worst case ratio can be made arbitrarily close to 4/3. For the restricted case where the setups are job-independent (sj = s), we demonstrate the existence of a polynomial time approximation scheme. The running times of our algorithms are linear in the number n of jobs and exponential in the reciprocal value of the desired precision.