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。以往文献主要集中于确定共流的调度顺序。为推进相关研究，我们通过确定流调度顺序来优化性能。所提出的近似算法在任意释放时间与零释放时间下分别达到$3$和$2+\frac{1}{LB}$的近似比，其中$LB$为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方法简化线性规划，从而减少变量数量。结果表明，对于$\epsilon>0$，该近似算法在任意释放时间与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此前在任意释放时间与零释放时间下分别达到的5和4的最佳近似比。