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This article focuses on the problem of coflow scheduling with precedence constraints in identical parallel networks, a well-known   $\mathcal {NP}$    NP-hard problem. Coflow is a relatively new network abstraction that characterizes communication patterns in data centers. When considering workload sizes and weights that are dependent on the network topology in the input instances, the proposed algorithm for the flow-level scheduling problem achieves an approximation ratio of   $O(\chi)$O  (  &#x03C7;  )where   $\chi$    &#x03C7;is the coflow number of the longest path in the directed acyclic graph (DAG). Additionally, when taking into account topology-dependent workload sizes, the algorithm achieves an approximation ratio of   $O(R\chi)$O  (  R  &#x03C7;  ), where   $R$    Rrepresents the ratio of maximum weight to minimum weight. For the coflow-level scheduling problem, the proposed algorithm achieves an approximation ratio of   $O(m\chi)$O  (  m  &#x03C7;  ), where   $m$    mis the number of network cores when considering workload sizes and weights that are topology-dependent. Moreover, when considering topology-dependent workload sizes, the algorithm achieves an approximation ratio of   $O(Rm\chi)$O  (  R  m  &#x03C7;  ). In the coflows of multi-stage job scheduling problem, the proposed algorithm achieves an approximation ratio of   $O(\chi)$O  (  &#x03C7;  ). 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.

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Efficient Approximation Algorithms for Scheduling Coflows With Precedence Constraints in Identical Parallel Networks to Minimize Weighted Completion Time