Approximation schemes for preemptive weighted flow time

(MATH) We present the first approximation schemes for minimizing weighted flow time on a single machine with preemption. Our first result is an algorithm that computes a (1+ε)-approximate solution for any instance of weighted flow time in <i>O</i>(<i>n</i><sup>O(ln <i>W</i> ln <i>P</i>/ε<sup>3</sup>)</sup>) time; here <i>P</i> is the ratio of maximum job processing time to minimum job processing time, and <i>W</i> is the ratio of maximum job weight to minimum job weight. This result directly gives a quasi-PTAS for weighted flow time when <i>P</i> and <i>W</i> are poly-bounded, and a PTAS when they are both <i>O</i>(1). We strengthen the former result to show that in order to get a quasi- PTAS it suffices to have just one of <i>P</i> and <i>W</i> to be poly-bounded. Our result provides strong evidence to the hypothesis that the weighted flow time problem has a PTAS. We note that the problem is strongly NP-hard even when <i>P</i> and <i>W</i> are <i>O</i>(1). We next consider two important special cases of weighted flow time, namely, when <i>P</i> is <i>O</i>(1) and <i>W</i> is arbitrary, and when the weight of a job is inverse of its processing time referred to as the stretch metric. For both of the above special cases we obtain a (1+ε)-approximation for any ε ρ 0 by using a randomized partitioning scheme to reduce an arbitrary instance to several instances all of which have <i>P</i> and <i>W</i> bounded by a constant that depends only on ε.

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