This paper addresses the challenging scheduling problem of coflows with release times, with the objective of minimizing the total weighted completion time. Previous literature has predominantly concentrated on establishing the scheduling order of coflows. In advancing this research, we contribute by optimizing performance through the determination of the flow scheduling order. The proposed approximation algorithm achieves approximation ratios of $3$ and $2+\frac{1}{LB}$ for arbitrary and zero release times, respectively, where $LB$ is the minimum lower bound of coflow completion time. To further improve time complexity, we streamline linear programming by employing interval-indexed relaxation, thereby reducing the number of variables. As a result, for $\epsilon>0$, the approximation algorithm achieves approximation ratios of $3 + \epsilon$ and $2 + \epsilon$ for arbitrary and zero release times, respectively. Notably, these advancements surpass the previously best-known approximation ratios of 5 and 4 for arbitrary and zero release times, respectively, as established by Shafiee and Ghaderi.

翻译：本文研究了具有释放时间的共流调度难题，目标是最小化总加权完成时间。现有文献主要集中于确定共流的调度顺序。为推进该研究，我们通过确定流调度顺序来优化性能，并以此作出贡献。所提出的逼近算法在任意释放时间和零释放时间下分别实现了$3$和$2+\frac{1}{LB}$的逼近比，其中$LB$是共流完成时间的最小下界。为进一步提升时间复杂度，我们采用区间索引松弛简化线性规划，从而减少了变量数量。因此，对于$\epsilon>0$，该逼近算法在任意释放时间和零释放时间下分别实现了$3 + \epsilon$和$2 + \epsilon$的逼近比。值得注意的是，这些进展超越了Shafiee和Ghaderi先前在任意释放时间和零释放时间下分别取得的5和4的最优逼近比。