Online Scheduling on Identical Machines using SRPT

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Due to its optimality on a single machine for the problem of minimizing average flow time, Shortest-Remaining-Processing-Time () appears to be the most natural algorithm to consider for the problem of minimizing average flow time on multiple identical machines. It is known that achieves the best possible competitive ratio on multiple machines up to a constant factor. Using resource augmentation, is known to achieve total flow time at most that of the optimal solution when given machines of speed 2- 1/m. Further, it is known that 's competitive ratio improves as the speed increases; is s-speed 1/s-competitive when s ≥ 2- 1/m. However, a gap has persisted in our understanding of. Before this work, the performance of was not known when is given (1+)-speed when 0 < < 1-1/m, even though it has been thought that is (1+)-speed O(1)-competitive for over a decade. Resolving this question was suggested in Open Problem 2.9 from the survey "Online Scheduling" by Pruhs, Sgall, and Torng PruhsST, and we answer the question in this paper. We show that is scalable on m identical machines. That is, we show is (1+)-speed O(1/)-competitive for >0. We complement this by showing that is (1+)-speed O(1/²)-competitive for the objective of minimizing the ℓₖ-norms of flow time on m identical machines. Both of our results rely on new potential functions that capture the structure of. Our results, combined with previous work, show that is the best possible online algorithm in essentially every aspect when migration is permissible.