In 2022, Chen et al. proposed an algorithm in \cite{main} that solves the min cost flow problem in $m^{1 + o(1)} \log U \log C$ time, where $m$ is the number of edges in the graph, $U$ is an upper bound on capacities and $C$ is an upper bound on costs. However, as far as the authors of \cite{main} know, no one has implemented their algorithm to date. In this paper, we discuss implementations of several key portions of the algorithm given in \cite{main}, including the justifications for specific implementation choices. For the portions of the algorithm that we do not implement, we provide stubs. We then go through the entire algorithm and calculate the $m^{o(1)}$ term more precisely. Finally, we conclude with potential directions for future work in this area.

翻译：2022年，Chen等人在文献\cite{main}中提出了一种算法，可在$m^{1 + o(1)} \log U \log C$时间内求解最小费用流问题，其中$m$为图的边数，$U$是容量的上界，$C$是费用的上界。然而，据文献\cite{main}作者所知，迄今为止尚未有人实现该算法。本文讨论了文献\cite{main}所给算法中若干关键部分的实现方案，包括具体实现选择的依据。对于未实现的部分，我们提供了桩模块。随后，我们通览整个算法并更精确地计算了$m^{o(1)}$项。最后，我们展望了该领域未来工作的潜在方向。