This paper addresses the scheduling problem of coflows in identical parallel networks, which is a well-known $NP$-hard problem. Coflow is a relatively new network abstraction used to characterize communication patterns in data centers. We consider both flow-level scheduling and coflow-level scheduling problems. In the flow-level scheduling problem, flows within a coflow can be transmitted through different network cores. However, in the coflow-level scheduling problem, flows within a coflow must be transmitted through the same network core. The key difference between these two problems lies in their scheduling granularity. Previous approaches relied on linear programming to solve the scheduling order. In this paper, we enhance the efficiency of solving by utilizing the primal-dual method. For the flow-level scheduling problem, we propose a $(6-\frac{2}{m})$-approximation algorithm with arbitrary release times and a $(5-\frac{2}{m})$-approximation algorithm without release time, where $m$ represents the number of network cores. Additionally, for the coflow-level scheduling problem, we introduce a $(4m+1)$-approximation algorithm with arbitrary release times and a $(4m)$-approximation algorithm without release time. To validate the effectiveness of our proposed algorithms, we conduct simulations using both synthetic and real traffic traces. The results demonstrate the superior performance of our algorithms compared to previous approach, emphasizing their practical utility.

翻译：本文研究了相同并行网络中的协同流调度问题，这是一个著名的$NP$-难问题。协同流是一种相对新颖的网络抽象概念，用于描述数据中心中的通信模式。我们同时考虑了流级调度和协同流级调度问题。在流级调度问题中，同一协同流内的流可以通过不同的网络核心传输。然而，在协同流级调度问题中，同一协同流内的流必须通过相同的网络核心传输。这两个问题的关键区别在于它们的调度粒度。以往的方法依赖线性规划来确定调度顺序。本文通过利用原始-对偶方法提高了求解效率。针对流级调度问题，我们提出了一个具有任意释放时间的$(6-\frac{2}{m})$-近似算法和一个无释放时间的$(5-\frac{2}{m})$-近似算法，其中$m$表示网络核心数量。此外，针对协同流级调度问题，我们引入了一个具有任意释放时间的$(4m+1)$-近似算法和一个无释放时间的$(4m)$-近似算法。为验证所提算法的有效性，我们使用合成流量数据和真实流量轨迹进行了仿真实验。结果表明，与以往方法相比，我们的算法具有更优的性能，凸显了其实用价值。