A competitive algorithm for minimizing weighted flow time on unrelatedmachines with speed augmentation

Abstract

We consider the online problem of scheduling jobs on unrelated machines so as to minimize the total weighted flow time. This problem has an unbounded competitive ratio even for very restricted settings. In this paper we show that if we allow the machines of the online algorithm to have more speed than those of the offline algorithm then we can get an O((1 + −1 ) 2 )-competitive algorithm.Our algorithm schedules jobs preemptively but without migration. However, we compare our solution to an offline algorithm which allows migration. Our analysis uses a potential function argument which can also be extended to give a simpler and better proof of the randomized immediate dispatch algorithm of Chekuri-Goel-Khanna-Kumar for minimizing average flow time on parallel machines.

Coordination mechanisms from (almost) all scheduling policies

“…We present two analysis techniques which yield somewhat different bounds -potential functions and dual fitting. Conceptually, our techniques are inspired by the elegant work on online scheduling [7,1,15] -the connection being that in both cases, we need to compare the decisions made by the optimal solution (that is non-strategic and omniscient) with the solution that arises due to the execution of the implemented policy. However, the similarity ends thereunlike online algorithms, in a coordination game, a job is selfish and cannot be forced to go to a specific machine and hence equilibrium state cannot be controlled by an algorithm.…”

“…When the jobs arrive online, however, no natural non-preemptive scheduling policy gives O(1) approximation to the weighted completion time, even on a single machine. This is particularly relevant since both our potential function and dual-fitting proofs are inspired by the frameworks developed for online scheduling problems [7,1]. Nevertheless, in Appendix C we present a general black box reduction from preemptive scheduling policies to non-preemptive scheduling policies that preserve the stretch within a factor of two.…”

“…Our potential function and dual-fitting techniques are inspired by the elegant framework developed for online scheduling in [7,1]. We note that their framework can be adapted to yield combinatorial scheduling policies for weighted completion time that are also O(1)-approximations.…”

“…It is straightforward to extend our analysis to incorporate nonnegative assignment costs. 7 Note that the left hand side of Equation 1 results from mixing two completely different scenarios S and S . Hence, it is not easy to relate this quantity with the right hand side, which consists of the weighted completion times under the two individual scenarios.…”

“…Note that the assignment vector and the strategy-profile remain the same across the two scenarios S and S (δ). 7 The dual fitting proof in Section 4 is presented in its full generality, and does not require this simplifying assumption.…”