Unrelated parallel machine scheduling with resource dependent processing times

We consider machine scheduling problems where jobs have to be processed on unrelated parallel machines in order to minimize the schedule makespan. The processing time of any job is dependent on the usage of a scarce resource that can be distributed over the jobs in process. The more of that resource is allocated to a job, the smaller its processing time. This model is a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos, the main di®erence lying in the fact that we assume that the resource is renewable, and not a total budget constraint. Using linear programming based rounding techniques, combined with a simple greedy algorithm, we derive the ¯rst constant-factor approximation algorithm for this problem. Our approach can be adapted to yield an approximation algorithm for the problem with dedicated machines, too. Moreover, we prove a non-approximability result for the problem with dedicated machines, and we derive a lower bound for the employed analysis.