A prominent problem in scheduling theory is the weighted flow time problem on one machine. We are given a machine and a set of jobs, each of them characterized by a processing time, a release time, and a weight. The goal is to find a (possibly preemptive) schedule for the jobs in order to minimize the sum of the weighted flow times, where the flow time of a job is the time between its release time and its completion time. It had been a longstanding important open question to find a polynomial time $O(1)$-approximation algorithm for the problem and this was resolved in a recent line of work. These algorithms are quite complicated and involve for example a reduction to (geometric) covering problems, dynamic programs to solve those, and LP-rounding methods to reduce the running time to a polynomial in the input size. In this paper, we present a much simpler $(6+\epsilon)$-approximation algorithm for the problem that does not use any of these reductions, but which works on the input jobs directly. It even generalizes directly to an $O(1)$-approximation algorithm for minimizing the $p$-norm of the jobs' flow times, for any $0 < p < \infty$ (the original problem setting corresponds to $p=1$). Prior to our work, for $p>1$ only a pseudopolynomial time $O(1)$-approximation algorithm was known for this variant, and no algorithm for $p<1$. For the same objective function, we present a very simple QPTAS for the setting of constantly many unrelated machines for $0 < p < \infty$ (and assuming quasi-polynomially bounded input data). It works in the cases with and without the possibility to migrate a job to a different machine. This is the first QPTAS for the problem if migrations are allowed, and it is arguably simpler than the known QPTAS for minimizing the weighted sum of the jobs' flow times without migration.

翻译：调度理论中的一个重要问题是单机上的加权流时间问题。我们有一台机器和一组作业，每个作业由处理时间、释放时间和权重刻画。目标是找到一个（可能可抢占的）调度方案，以最小化加权流时间之和，其中作业的流时间是指其释放时间到完成时间之间的时间间隔。长期以来，为这一问题寻找多项式时间 $O(1)$ 近似算法一直是一个重要的未解难题，而这一问题在最近的一系列工作中得到了解决。这些算法相当复杂，例如涉及归约到（几何）覆盖问题、用于求解这些问题的动态规划，以及用于将运行时间降至输入规模多项式的 LP 舍入方法。在本文中，我们针对该问题提出了一种更简单的 $(6+\epsilon)$ 近似算法，该算法不使用任何上述归约，而是直接处理输入作业。它甚至能直接推广到适用于任意 $0 < p < \infty$ 的 $O(1)$ 近似算法，用于最小化作业流时间的 $p$-范数（原始问题设置对应于 $p=1$）。在我们工作之前，对于 $p>1$，仅已知该变体的一种伪多项式时间 $O(1)$ 近似算法，而对于 $p<1$ 则没有算法。针对相同的目标函数，我们针对具有常数台不相关机器（并假设输入数据为准多项式有界）的设置，提出了一种非常简单的 QPTAS，适用于 $0 < p < \infty$。它适用于允许和不允许将作业迁移到不同机器的情况。这是该问题在允许迁移情况下的首个 QPTAS，并且可以说它比已知的用于最小化无迁移的作业流时间加权和的 QPTAS 更简单。