SIAM Journal on Computing, Ahead of Print.
In the weighted flow-time problem on a single machine, we are given a set of $n$ jobs, where each job has a processing requirement $p_j$, release date $r_j$, and weight $w_j$. The goal is to find a preemptive schedule which minimizes the sum of weighted flow-time of jobs, where the flow-time of a job is the difference between its completion time and its released date. We give the first pseudo-polynomial time constant approximation algorithm for this problem. The algorithm also extends directly to the problem of minimizing the $\ell_p$ norm of weighted flow-times. The running time of our algorithm is polynomial in $n$, the number of jobs, and $P$, which is the ratio of the largest to the smallest processing requirement of a job. Our algorithm relies on a novel reduction of this problem to a generalization of the multicut problem on trees, which we call the Demand MultiCut problem. Even though we do not give a constant factor approximation algorithm for the Demand MultiCut problem on trees, we show that the specific instances of Demand MultiCut obtained by reduction from weighted flow-time problem instances have more structure in them, and we are able to employ techniques based on dynamic programming. Our dynamic programming algorithm relies on showing that there are near optimal solutions which have nice smoothness properties, and we exploit these properties to reduce the size of the dynamic programming table.

中文翻译：

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伪多项式时间内单机加权流时间的常数因子近似算法

《 SIAM计算杂志》，预印本。
在单台机器上的加权流动时间问题中，我们得到了一组$ n $个作业，其中每个作业都有处理要求$ p_j $，发布日期$ r_j $和权重$ w_j $。目标是找到一种抢占式计划，以最大程度地减少作业的加权流动时间总和，其中作业的流动时间是其完成时间与发布日期之间的差。针对该问题，我们给出了第一个伪多项式时间常数近似算法。该算法还直接扩展到最小化加权流时间的\ ell_p $范数的问题。我们的算法的运行时间是多项式，单位为$ n $，作业数和$ P $，这是作业的最大处理需求与最小处理需求的比率。我们的算法依靠对该问题的新颖化简为对树上多割问题的推广，我们称之为需求多切问题。即使我们没有为树上的Demand MultiCut问题提供恒定因子近似算法，我们也表明，通过从加权流动时间问题实例中还原而获得的Demand MultiCut的特定实例在其中具有更多的结构，并且能够采用基于动态编程的技术。我们的动态规划算法依赖于显示具有最佳平滑性的近乎最优解，并且我们利用这些属性来减小动态规划表的大小。我们表明，通过从加权流动时间问题实例中进行还原而获得的Demand MultiCut特定实例具有更多的结构，并且我们能够采用基于动态编程的技术。我们的动态规划算法依赖于显示具有最佳平滑性的近最优解，我们利用这些属性来减小动态规划表的大小。我们表明，通过从加权流动时间问题实例中进行还原而获得的Demand MultiCut特定实例具有更多的结构，并且我们能够采用基于动态编程的技术。我们的动态规划算法依赖于显示具有最佳平滑性的近最优解，我们利用这些属性来减小动态规划表的大小。