  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.
