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Coflow is a network abstraction used to represent communication patterns indata centers. The coflow scheduling problem encountered in large data centersis a challenging $\mathcal{NP}$-hard problem. This paper tackles the schedulingproblem of coflows with release times in heterogeneous parallel networks, whichfeature an architecture consisting of multiple network cores running inparallel. Two polynomial-time approximation algorithms are presented in thispaper, designed to minimize the total weighted completion time and makespan inheterogeneous parallel networks, respectively. For any given $\epsilon>0$, ourproposed approximation algorithm for minimizing the total weighted completiontime achieves approximation ratios of $3 + \epsilon$ and $2 + \epsilon$ in thecases of arbitrary and zero release times, respectively. Additionally, weintroduce an approximation algorithm for minimizing the makespan, achieving anapproximation ratio of $2 + \epsilon$ for $\epsilon>0$. Notably, theseadvancements surpass the previously best-known approximation ratio of $O(\logm/ \log \log m)$ for both minimizing the total weighted completion time andmakespan. This result also improves upon the previous approximation ratios of$6-\frac{2}{m}$ and $5-\frac{2}{m}$ for arbitrary and zero release times,respectively, in identical parallel networks.

Improved Approximation Coflows Scheduling Algorithms for Minimizing the  Total Weighted Completion Time and Makespan in Heterogeneous Parallel  Networks