Machine scheduling with resource dependent processing times

We consider several parallel machine scheduling settings with the objective to minimize the schedule makespan. The most general of these settings is unrelated parallel machine scheduling. We assume that, in addition to its machine dependence, the processing time of any job is dependent on the usage of a scarce renewable resource. A given amount of that resource, e.g. workers, can be distributed over the jobs in process at any time, and the more of that resource is allocated to a job, the smaller is its processing time. This model generalizes classical machine scheduling problems, adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos. On the basis of integer programming formulations for relaxations of the respective problems, we use LP rounding techniques to allocate resources to jobs, and to assign jobs to machines. Combined with Graham’s list scheduling, we thus prove the existence of constant factor approximation algorithms. Our performance guarantee is (4 + 2 √ 2) ≈ 6.83 for the most general case of unrelated parallel machine scheduling. We improve this bound for two special cases, namely to (3 + 2 √ 2) ≈ 5.83 whenever the jobs are assigned to machines beforehand, and to (5 + ε) whenever the processing times do not depend on the machine. Moreover, we discuss tightness of the relaxations, and derive inapproximability results.