We consider the precedence-constrained scheduling problem to minimize the total weighted completion time. For a single machine several $2$-approximation algorithms are known, which are based on linear programming and network flows. We show that the same ratio is achieved by a simple weighted round-robin rule. Moreover, for preemptive scheduling on identical parallel machines, we give a strongly polynomial $3$-approximation, which computes processing rates by solving a sequence of parametric flow problems. This matches the best known constant performance guarantee, previously attained only by a weakly polynomial LP-based algorithm. Our algorithms are both also applicable in non-clairvoyant scheduling, where processing times are initially unknown. In this setting, our performance guarantees improve upon the best competitive ratio of $8$ known so far.

翻译：我们考虑优先约束调度问题，旨在最小化总加权完成时间。对于单机环境，已有若干基于线性规划和网络流的2-近似算法。我们证明，一种简单的加权轮询规则同样能达到该近似比。此外，对于同构并行机上的抢占式调度，我们提出了一种强多项式时间的3-近似算法，该算法通过求解一系列参数化流问题来计算处理速率。该结果与之前仅通过弱多项式时间基于线性规划的算法所取得的最佳常数性能保证相匹配。我们的两种算法同样适用于非预见性调度场景，其中处理时间初始未知。在该场景下，我们的性能保证优于目前已知的最佳竞争比8。