Unrelated Parallel Machine Scheduling with Resource Dependent Processing Times

We consider unrelated parallel machine scheduling problems with the objective to minimize the schedule makespan. In addition to its machine-dependence, the processing time of any job is also dependent on the usage of a scarce renewable resource. An amount of k units of that resource, e.g. workers, can be distributed over the jobs in process, and the more of that resource is allocated to a job, the smaller its processing time. The model generalizes the classical unrelated machine scheduling problem, adding a resource-time tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos, the difference lying in the fact the resource is renewable and not a total budget constraint. We use a two-phased LP rounding technique to assign resources to jobs and jobs to machines. Combined with Graham's list scheduling, we thus prove the existence of a (4+2√2)-approximation algorithm. We show how our approach can be adapted to scheduling problems with dedicated machines as well, with an improvement of the performance bound to (3+2√2). Moreover, we derive a lower bound of 2 for the employed LP-based analysis, and we prove a (3/2)-inapproximability result.