Preemptive Online Scheduling: Optimal Algorithms for All Speeds

Our main result is an optimal online algorithm for preemptive scheduling on uniformly related machines with the objective to minimize makespan. The algorithm is determin-istic, yet it is optimal even among all randomized algorithms. In addition, it is optimal for any fixed combination of speeds of the machines, and thus our results subsume all the previous work on various special cases. Together with a new lower bound it follows that the overall competitive ratio of this optimal algorithm is between 2.054 and e ≈ 2.718. We also give a complete analysis of the competitive ratio for three machines.