This paper focuses on the problem of coflow scheduling with precedence constraints in identical parallel networks, which is a well-known $\mathcal{NP}$-hard problem. Coflow is a relatively new network abstraction used to characterize communication patterns in data centers. Both flow-level scheduling and coflow-level scheduling problems are examined, with the key distinction being the scheduling granularity. The proposed algorithm effectively determines the scheduling order of coflows by employing the primal-dual method. When considering workload sizes and weights that are dependent on the network topology in the input instances, our proposed algorithm for the flow-level scheduling problem achieves an approximation ratio of $O(\chi)$ where $\chi$ is the coflow number of the longest path in the directed acyclic graph (DAG). Additionally, when taking into account workload sizes that are topology-dependent, the algorithm achieves an approximation ratio of $O(R\chi)$, where $R$ represents the ratio of maximum weight to minimum weight. For the coflow-level scheduling problem, the proposed algorithm achieves an approximation ratio of $O(m\chi)$, where $m$ is the number of network cores, when considering workload sizes and weights that are topology-dependent. Moreover, when considering workload sizes that are topology-dependent, the algorithm achieves an approximation ratio of $O(Rm\chi)$. In the coflows of multi-stage job scheduling problem, the proposed algorithm achieves an approximation ratio of $O(\chi)$. Although our theoretical results are based on a limited set of input instances, experimental findings show that the results for general input instances outperform the theoretical results, thereby demonstrating the effectiveness and practicality of the proposed algorithm.

翻译：本文研究同构并行网络中具有优先约束的协同流调度问题，该问题是一个经典的$\mathcal{NP}$-难问题。协同流是一种相对较新的网络抽象概念，用于表征数据中心中的通信模式。本文分别考察了流级调度与协同流级调度问题，其核心区别在于调度粒度。所提算法通过采用原始-对偶方法有效确定协同流的调度顺序。当输入实例中工作负载大小和权重依赖于网络拓扑时，针对流级调度问题，所提算法达到了$O(\chi)$的近似比，其中$\chi$为有向无环图（DAG）中最长路径上的协同流数量。此外，当考虑依赖于拓扑的工作负载大小时，该算法实现了$O(R\chi)$的近似比，其中$R$表示最大权重与最小权重之比。针对协同流级调度问题，当考虑依赖于拓扑的工作负载大小和权重时，所提算法达到了$O(m\chi)$的近似比，其中$m$为网络核心数量。进一步地，当考虑依赖于拓扑的工作负载大小时，算法实现了$O(Rm\chi)$的近似比。在多阶段作业调度问题的协同流场景中，所提算法达到了$O(\chi)$的近似比。尽管理论结果基于有限输入实例，但实验结果表明，在通用输入实例上的性能优于理论结果，从而验证了所提算法的有效性与实用性。