Summary: 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 scarce 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 dependence of processing times on the amount of resources is linear for any job. The objective is to find a resource allocation and a schedule that minimizes the makespan. Utilizing an integer quadratic programming relaxation, we show how to obtain a \((3+\varepsilon)\)-approximation algorithm for that problem, for any \(\varepsilon > 0\). This generalizes and improves previous results, respectively. Our approach relies on a fully polynomial time approximation scheme to solve the quadratic programming relaxation. This result is interesting in itself, because the underlying quadratic program is NP-hard to solve. We also derive lower bounds, and discuss further generalizations of the results.
For the entire collection see [Zbl 1097.68003].