  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.
