Optimal preemptive scheduling for general target functions

We study the problem of optimal preemptive scheduling with respect to a general target function. Given n jobs with associated weights and m less than or equal to n uniformly related machines, one aims at scheduling the jobs to the machines, allowing preemptions but forbidding parallelization, so that a given target function of the loads on each machine is minimized. This problem was studied in the past in the case of the makespan. Gonzalez and Sahni [7] and later Shachnai, Tamir and Woeginger [12] devised a polynomial algorithm that outputs an optimal schedule for which the number of preemptions is at most 2(m - 1). We extend their ideas for general symmetric, convex and monotone target functions. This general approach enables us to distill the underlying principles on which the optimal makespan algorithm is based. More specifically, the general approach enables us to identify between the optimal scheduling problem and a corresponding problem of mathematical programming. This, in turn, allows us to devise a single algorithm that is suitable for a wide array of target functions, where the only difference between one target function and another is manifested through the corresponding mathematical programming problem.