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：本文研究带有释放时间的Coflow调度难题，旨在最小化总加权完成时间。现有文献主要集中于确定Coflow的调度顺序。在此基础上，我们通过确定流调度顺序来优化性能，从而推动该研究进展。所提出的近似算法在任意释放时间和零释放时间下分别达到$3$和$2+\frac{1}{LB}$的近似比，其中$LB$是Coflow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，从而减少变量数量。因此，对于$\epsilon>0$，该近似算法在任意释放时间和零释放时间下分别达到$3 + \epsilon$和$2 + \epsilon$的近似比。值得注意的是，这些进展超越了Shafiee和Ghaderi先前针对任意释放时间和零释放时间所建立的最佳已知近似比5和4。