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We consider a scheduling problem on unrelated parallel machines with the objective to minimize the makespan. In addition to its machine dependence, the processing time of any job is dependent on the usage of a scarce renewable resource, e.g. workers. A given amount of that resource can be distributed over the jobs in process at any time. The more of the resource is allocated to a job, the smaller is its processing time. This model generalizes the classical unrelated parallel machine scheduling problem by adding a time-resource tradeoff. It is also a natural variant of a generalized assignment problem studied by Shmoys and Tardos. On the basis of an integer linear programming formulation for (a relaxation of) the problem, we adopt a randomized LP rounding technique from Kumar et al. (FOCS 2005) in order to obtain a deterministic, integral LP solution that is close to optimum. We show how this rounding procedure can be used to derive a deterministic 3.75-approximation algorithm for the scheduling problem. This improves upon previous results, namely a deterministic 6.83-approximation, and a randomized 4-approximation. The improvement is due to the better LP rounding and a new scheduling algorithm that can be viewed as a restricted version of the harmonic algorithm for bin packing.

We consider the problem of pricing items in order to maximize the revenue obtainable from a set of single minded customers. We relate the tractability of the problem to structural properties of customers' valuations: the problem admits an effcient approximation algorithm, parameterized along the inhomogeneity of the valuations.

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 + e) -approximation algorithm for that problem, for any e > 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.

We consider unrelated parallel machine scheduling problems with the objective to minimize the schedule makespan. In addition to its machine-dependence, the processing time of any job is also dependent on the usage of a scarce renewable resource. An amount of k units of that resource, e.g. workers, can be distributed over the jobs in process, and the more of that resource is allocated to a job, the smaller its processing time. The model generalizes the classical unrelated machine scheduling problem, adding a resource-time tradeoff. It is also a natural variant of a generalized assignment problem studied previously by Shmoys and Tardos, the difference lying in the fact the resource is renewable and not a total budget constraint. We use a two-phased LP rounding technique to assign resources to jobs and jobs to machines. Combined with Graham's list scheduling, we thus prove the existence of a $(4+2\sqrt{2})$-approximation algorithm. We show how our approach can be adapted to scheduling problems with dedicated machines as well, with an improvement of the performance bound to $(3+2\sqrt{2})$. Moreover, we derive a lower bound of 2 for the employed LP-based analysis, and we prove a (3/2)-inapproximability result.