Optimal Preemptive Scheduling for General Target Functions

Abstract

We study the problem of optimal preemptive scheduling with respect to a general target function. Given n jobs with associated weights and m ≤ 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 [6] 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.

Preemptive scheduling on uniformly related machines: minimizing the sum of the largest pair of job completion times

“…Shachnai et al (2005) generalized the above algorithm for jobs of limited splitting constraints, that is, a job cannot be J Sched split into an arbitrary number of parts, but only to a limited number of parts, depending on the job. Epstein and Tassa (2006) obtained similar results for a wide class of goal functions, including minimization of the p norm of the vector of machine completion times. This vector is of length m, with one component for each machine representing its completion time, such that component i represents the completion time of machine i.…”

“…The algorithm InTime receives a parameter T > 0 (in addition to a set of machines and a set of jobs), and creates a schedule of the input jobs on the input machines such that all jobs are completed no later than time T , if this is possible, that is, if the optimal makespan for the given input does not exceed T . This algorithm is similar to those of Gonzales and Sahni (1978), Shachnai et al (2005), Epstein and Tassa (2006), but it can act on an unsorted list of jobs. Its running time (for m machines and n jobs) is O(n + m log m) if m ≥ n, and therefore, using our notation, its running time is O(m + n +m logm).…”

“…The next lemma has been used in previous work, and the bounds for the minimum makespan are based on similar arguments (see e.g., Epstein and Tassa 2006).…”