Scheduling parallel jobs with monotone speedup

We consider a scheduling problem where a set of jobs is a-priori distributed over parallel machines. The processing time of any job is dependent on the usage of a discrete renewable resource, e.g. personnel. An amount of k units of that resource can be allocated to the jobs at any time, and the more of that resource is allocated to a job, the smaller its processing time. The objective is to find a resource allocation and a schedule that minimizes the makespan. In contrast to previous approaches, we explicitly allow for succinctly encodable time-resource tradeoff functions. This assumption calls for mathematical programming techniques other than those that have been used before. Utilizing a (nonlinear) integer mathematical program, we obtain the first polynomial time approximation algorithm for the scheduling problem, with performance bound (3 + e) for any e > 0. Our approach relies on a fully polynomial time approximation scheme to solve the mathematical programming relaxation. This result is interesting in itself, because we also show that the underlying mathematical program is NP-hard to solve. We also derive lower bounds for the approximation.