As a prominent network abstraction, coflow models efficiently capture communication patterns in data centers. Since coflow scheduling in large-scale data centers is $\mathcal{NP}$-hard, the existing literature has predominantly focused on limited environments with $m=2$ network cores, relying on flow splitting, which introduces substantial operational overhead. Crucially, no approximation algorithm with provable performance guarantees has been proposed for the more practical, non-splitting coflow scheduling problem, even for the $m=2$ case, let alone for general hybrid architectures. To bridge this critical gap, this paper investigates the non-splitting problem within a hybrid, heterogeneous parallel network featuring multiple network cores ($m \ge 2$) composed of Electronic Packet Switches (EPS), not-all-stop Optical Circuit Switches (OCS), and all-stop OCS. We propose a unified polynomial-time approximation algorithm that minimizes the makespan across this hybrid environment without incurring any splitting overhead. Let $τ$ denote the maximum flow degree across all ports in the network, $N$ be the number of input/output ports, and $m$ be the number of network cores. In pure EPS environments, the algorithm achieves an approximation guarantee of $\min\left\{τ, m\right\}$. For pure not-all-stop and pure all-stop OCS environments, the guaranteed ratios are $2\min\left\{τ, m\right\}$ and $2\min\left\{2τ-1, m+τ-1\right\}$, respectively. Notably, when specialized to the $m=2$ setting, our algorithm achieves constant bounds of $2$ and $4$ for pure EPS, and pure not-all-stop OCS, respectively, and $2τ+2$ for pure all-stop OCS. By leveraging these constituent bounds, we prove that the overall performance guarantee in the hybrid architecture is upper-bounded by the least-performing switch architecture in the network.

翻译：作为显著的网络抽象模型，共流高效地捕获了数据中心中的通信模式。由于大规模数据中心中的共流调度问题是$\mathcal{NP}$-难的，现有文献主要集中于有限环境（即$m=2$个网络核心），并依赖于引入显著操作开销的流量拆分。关键的是，对于更实用的非拆分共流调度问题，即使对于$m=2$的情况，也尚未提出任何具有可证明性能保证的近似算法，更不用说针对通用混合架构了。为填补这一关键空白，本文研究了混合异构并行网络中的非拆分问题，该网络包含多个网络核心（$m \ge 2$），由电子分组交换机（EPS）、非全停光电路交换机（OCS）和全停OCS组成。我们提出了一种统一的多项式时间近似算法，该算法在混合环境中最小化完工时间，且不产生任何拆分开销。设$τ$表示网络中所有端口上的最大流度，$N$为输入/输出端口数，$m$为网络核心数。在纯EPS环境中，该算法达到$\min\left\{τ, m\right\}$的近似比。对于纯非全停OCS和纯全停OCS环境，保证比分别为$2\min\left\{τ, m\right\}$和$2\min\left\{2τ-1, m+τ-1\right\}$。值得注意的是，当特化为$m=2$设置时，我们的算法在纯EPS和纯非全停OCS下分别达到常数界$2$和$4$，在纯全停OCS下达到$2τ+2$。通过利用这些构成界，我们证明混合架构中的整体性能保证受限于网络中性能最差的交换架构。