  Coflow is a network abstraction used to represent communication patterns in
data centers. The coflow scheduling problem encountered in large data centers
is a challenging $\mathcal{NP}$-hard problem. This paper tackles the scheduling
problem of coflows with release times in heterogeneous parallel networks, which
feature an architecture consisting of multiple network cores running in
parallel. Two polynomial-time approximation algorithms are presented in this
paper, designed to minimize the total weighted completion time and makespan in
heterogeneous parallel networks, respectively. For any given $\epsilon>0$, our
proposed approximation algorithm for minimizing the total weighted completion
time achieves approximation ratios of $3 + \epsilon$ and $2 + \epsilon$ in the
cases of arbitrary and zero release times, respectively. Additionally, we
introduce an approximation algorithm for minimizing the makespan, achieving an
approximation ratio of $2 + \epsilon$ for $\epsilon>0$. Notably, these
advancements surpass the previously best-known approximation ratio of $O(\log
m/ \log \log m)$ for both minimizing the total weighted completion time and
makespan. This result also improves upon the previous approximation ratios of
$6-\frac{2}{m}$ and $5-\frac{2}{m}$ for arbitrary and zero release times,
respectively, in identical parallel networks.
