The Power of Preemption on Unrelated Machines and Applications to Scheduling Orders

Scheduling jobs on unrelated parallel machines so as to minimize makespan is one of the basic problems in the area of machine scheduling. In the first part of the paper, we prove that the power of preemption, i.e., the worst-case ratio between the makespan of an optimal nonpreemptive and that of an optimal preemptive schedule, is at least 4. This matches the upper bound proposed in Lin and Vitter [Lin, J.-H., J. S. Vitter. 1992. ˜-approximations with minimum packing constraint violation. Proc. 24th Annual ACM Sympos. Theory of Comput. (STOC), ACM, New York, 771–782] two decades ago. In the second part of the paper, we consider the more general setting in which orders, consisting of several jobs, have to be processed on unrelated parallel machines so as to minimize the sum of weighted completion times of the orders. We obtain the first constant factor approximation algorithms for the preemptive and nonpreemptive cases, improving and extending a recent result by Leung et al. [Leung, J., H. Li, M. Pinedo, J. Zhang. 2007. Minimizing total weighted completion time when scheduling orders in a flexible environment with uniform machines. Inform. Processing Lett. 103 119–129]. Finally, we study this problem in a parallel machine environment, obtaining a polynomial-time approximation scheme for several special cases.