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This paper addresses the challenging scheduling problem of coflows withrelease times, with the objective of minimizing the total weighted completiontime. Previous literature has predominantly concentrated on establishing thescheduling order of coflows. In advancing this research, we contribute byoptimizing 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 timecomplexity, we streamline linear programming by employing interval-indexedrelaxation, 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-knownapproximation ratios of 5 and 4 for arbitrary and zero release times,respectively, as established by Shafiee and Ghaderi.

Improved Approximation Algorithms for Minimizing the Total Weighted Completion Time of Coflows