SIAM Journal on Computing, Ahead of Print.
We study approximation algorithms for problems of scheduling precedence constrained jobs with the objective of minimizing total weighted completion time, in identical and related machine models. We give algorithms that improve upon many previous 15- to 20-year-old state-of-the-art results. A major theme in these results is the use of time-indexed linear programming relaxations, which are quite natural for their respective problems. We also consider the scheduling problem of minimizing total weighted completion time on unrelated machines. The recent breakthrough result of [N. Bansal, A. Srinivasan, and O. Svensson, in Proceedings of the 48th Annual ACM Symposium on Theory of Computing, ACM, 2016, pp. 156--167] gave a (1.5-c)-approximation for the problem, based on a two-round lift-and-project of the SDP relaxation for the problem. Our main result is that a (1.5 - c)-approximation can also be achieved using a natural and considerably simpler time-indexed linear programming relaxation for the problem. We hope this relaxation can provide new insights into the problem.

中文翻译：

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通过时间索引线性规划松弛来计划以最小化总加权完成时间

《 SIAM计算杂志》，预印本。
我们研究在相同和相关机器模型中用于调度优先级受限作业的问题的近似算法，目的是最大程度地减少总加权完成时间。我们提供的算法可以改善许多以前的15到20年历史的最新结果。这些结果的主要主题是使用时间索引线性规划松弛，这对于它们各自的问题而言是很自然的。我们还考虑了使不相关机器上的总加权完成时间最小化的调度问题。[N. Bansal，A。Srinivasan和O. Svensson，在第48届ACM计算理论年度研讨会论文集，ACM，2016，第156--167页]中给出了该问题的（1.5-c）近似值，基于针对该问题的SDP放宽的两轮提升和项目。我们的主要结果是，使用该问题的自然且相当简单的时间索引线性规划松弛，也可以实现（1.5-c）逼近。我们希望这种放松可以为问题提供新的见解。