Unrelated Parallel Machine Scheduling with Resource Dependent Processing Times

We consider unrelated machine scheduling problems with the objective to minimize the schedule makespan. In addition to its machinedependence, 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 1.5-inapproximability result.