Preemptive and non-preemptive scheduling on two unrelated parallel machines

In this paper, for the problem of minimizing the makespan on two unrelated parallel machines we compare the quality of preemptive and non-preemptive schedules. It is known that there exists an optimal preemptive schedule with at most two preemptions. We show that the power of preemption, i.e., the ratio of the makespan computed for the best non-preemptive schedule to the makespan of the optimal preemptive schedule is at most 3/2. We also show that the ratio of the makespan computed for the best schedule with at most one preemption to the makespan of the optimal preemptive schedule is at most 9/8. For both models, we present polynomial-time algorithms that find schedules of the required quality. The established bounds match those previously known for a less general problem with two uniform machines. We have found one point of difference between the uniform and unrelated machines: if an optimal preemptive schedule contains exactly one preemption then the ratio of the makespan computed for the best non-preemptive schedule to the makespan of the optimal preemptive schedule is at most 4/3 if the two machines are uniform and remains 3/2 if the machines are unrelated.