We consider the CONGEST model on a network with $n$ nodes, $m$ edges, diameter $D$, and integer costs and capacities bounded by $\text{poly} n$. In this paper, we show how to find an exact solution to the minimum cost flow problem in $n^{1/2+o(1)}(\sqrt{n}+D)$ rounds, improving the state of the art algorithm with running time $m^{3/7+o(1)}(\sqrt nD^{1/4}+D)$ [Forster et al. FOCS 2021], which only holds for the special case of unit capacity graphs. For certain graphs, we achieve even better results. In particular, for planar graphs, expander graphs, $n^{o(1)}$-genus graphs, $n^{o(1)}$-treewidth graphs, and excluded-minor graphs our algorithm takes $n^{1/2+o(1)}D$ rounds. We obtain this result by combining recent results on Laplacian solvers in the CONGEST model [Forster et al. FOCS 2021, Anagnostides et al. DISC 2022] with a CONGEST implementation of the LP solver of Lee and Sidford [FOCS 2014], and finally show that we can round the approximate solution to an exact solution. Our algorithm solves certain linear programs, that generalize minimum cost flow, up to additive error $\epsilon$ in $n^{1/2+o(1)}(\sqrt{n}+D)\log^3 (1/\epsilon)$ rounds.

翻译：我们考虑一个具有$n$个节点、$m$条边、直径$D$以及成本和容量均为$\text{poly} n$有界整数的网络上的CONGEST模型。本文展示了如何在$n^{1/2+o(1)}(\sqrt{n}+D)$轮内找到最小费用流问题的精确解，改进了此前仅适用于单位容量图特例的运行时间为$m^{3/7+o(1)}(\sqrt nD^{1/4}+D)$的最优算法 [Forster et al. FOCS 2021]。对于某些特定图类，我们实现了更优的结果。特别地，对于平面图、扩张图、$n^{o(1)}$-亏格图、$n^{o(1)}$-树宽图以及排除子式图，我们的算法仅需$n^{1/2+o(1)}D$轮。我们通过将CONGEST模型中拉普拉斯求解器的最新成果 [Forster et al. FOCS 2021, Anagnostides et al. DISC 2022] 与Lee和Sidford [FOCS 2014]的线性规划求解器的CONGEST实现相结合，并最终证明可将近似解舍入为精确解，从而获得上述结果。我们的算法可求解泛化最小费用流问题的特定线性规划，在$n^{1/2+o(1)}(\sqrt{n}+D)\log^3 (1/\epsilon)$轮内达到$\epsilon$的加性误差。