An online scalable algorithm for minimizing lk-norms of weighted flow time on unrelated machines

We consider the problem of scheduling jobs that arrive online in the <i>unrelated</i> machine model to minimize <i>l</i><sub><i>k</i></sub> norms of weighted flowtime. In the unrelated setting, the processing time and weight of a job depends on the machine it is assigned to, and it is perhaps the most general machine model considered in scheduling literature. Chadha et al. [10] obtained a recent breakthrough result in obtaining the first non-trivial algorithm for minimizing weighted flowtime (that is, the <i>l</i><sub>1</sub> norm) in this very general setting via a novel potential function based analysis. They described a simple algorithm and showed that for any ε > 0 it is (1 + ε)-speed <i>O</i>(1/ε<sup>2</sup>)-competitive (a scalable algorithm). In this paper we give the first non-trivial and scalable algorithm for minimizing <i>l</i><sub><i>k</i></sub> norms of weighted flowtime in the unrelated machine model; for any ε > 0, the algorithm is <i>O</i>(<i>k</i>/ε<sup>2+2/<i>k</i></sup>)-competitive. The algorithm is immediate-dispatch and non-migratory. Our result is of both practical and theoretical interest. Scheduling to minimize <i>l</i><sub><i>k</i></sub> norms of flowtime for some small <i>k</i> > 1 has been shown to balance total response time and fairness, which is desirable in practice. On the theoretical side, <i>l</i><sub><i>k</i></sub> norms for <i>k</i> > 1 pose substantial technical hurdles when compared to when <i>k</i> = 1 even for the single machine case. Our work develops a novel potential function as well as several tools that can be used to lower bound the optimal solution.

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