Preemptive Bicriteria Scheduling for Parallel Jobs: O -line and On-line Algorithms

We introduce general algorithms for oo-line and on-line scheduling of parallel and independent jobs on a multiprocessor system. In the oo-line case we consider two models by allowing and disallowing release dates. Following Hall, Schulz, Shmoys, and Wein 11], we achieve a constant approximation factor for both minimizing weighted sum of the job completion times and makespan objectives. Following Schulz and Skutella 17] 18] we achieve a smaller randomized constant factor for weighted sum of completion times, and a small constant factor for makespan. In both techniques we use an optimal solution to a linear programming relaxation in order to guide a task permutation as an input to a list scheduling 8] with preemption rule. Our on-line technique gives constant factors for both the (weighted) sum of completion times and makespan objectives, and is a kind of generalization to the one-machine preemptive relaxation introduced by Chekuri, Motwani, Natarajan, and Stein 3]. To the best of our knowledge our on-line algorithms are the rst known for the problems under consideration.