Scheduling jobs with time-resource tradeoff via nonlinear programming

We consider a scheduling problem where 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. We explicitly allow for succinctly encodable time-resource tradeoff functions, which 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 $(3 + \varepsilon)$ for any $\varepsilon > 0$. Our approach relies on a fully polynomial time approximation scheme to solve the nonlinear mathematical programming relaxation. We also derive lower bounds for the approximation.