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.

翻译：Coflow是一种用于表示数据中心通信模式的网络抽象。大型数据中心中遇到的Coflow调度问题是一个具有挑战性的$\mathcal{NP}$-hard问题。本文研究了异构并行网络中具有释放时间的Coflow调度问题，该网络架构由多个并行运行的网络核心组成。本文提出了两种多项式时间近似算法，分别用于最小化异构并行网络中的总加权完成时间和完工时间。对于任意给定的$\epsilon>0$，我们在最小化总加权完成时间的近似算法中，在任意释放时间和零释放时间情况下分别达到了$3+\epsilon$和$2+\epsilon$的近似比。此外，我们引入了一种最小化完工时间的近似算法，对于$\epsilon>0$实现了$2+\epsilon$的近似比。值得注意的是，这些进展在最小化总加权完成时间和完工时间两方面均超越了此前最优的$O(\log m/\log\log m)$近似比。这一结果也改进了同构并行网络中任意释放时间和零释放时间情况下分别达到$6-\frac{2}{m}$和$5-\frac{2}{m}$的先前近似比。