We present an algorithm for min-cost flow in graphs with $n$ vertices and $m$ edges, given a tree decomposition of width $\tau$ and size $S$, and polynomially bounded, integral edge capacities and costs, running in $\widetilde{O}(m\sqrt{\tau} + S)$ time. This improves upon the previous fastest algorithm in this setting achieved by the bounded-treewidth linear program solver by [Dong-Lee-Ye,21] and [Gu-Song,22], which runs in $\widetilde{O}(m \tau^{(\omega+1)/2})$ time, where $\omega \approx 2.37$ is the matrix multiplication exponent. Our approach leverages recent advances in structured linear program solvers and robust interior point methods (IPM). For general graphs where treewidth is trivially bounded by $n$, the algorithm runs in $\widetilde{O}(m \sqrt n)$ time, which is the best-known result without using the Lee-Sidford barrier or $\ell_1$ IPM, demonstrating the surprising power of robust interior point methods. As a corollary, we obtain a $\widetilde{O}(\operatorname{tw}^3 \cdot m)$ time algorithm to compute a tree decomposition of width $O(\operatorname{tw}\cdot \log(n))$, given a graph with $m$ edges.

翻译：我们提出了一种在具有$n$个顶点和$m$条边的图中求解最小费用流的算法，给定宽度为$\tau$、规模为$S$的树分解，以及多项式有界的整数边容量和费用，该算法的运行时间为$\widetilde{O}(m\sqrt{\tau} + S)$。这改进了此前在此设置下由[Dong-Lee-Ye,21]和[Gu-Song,22]的有界树宽线性规划求解器实现的最快算法，其运行时间为$\widetilde{O}(m \tau^{(\omega+1)/2})$，其中$\omega \approx 2.37$是矩阵乘法指数。我们的方法利用了结构化线性规划求解器和鲁棒内点法（IPM）的最新进展。对于树宽平凡有界于$n$的一般图，该算法的运行时间为$\widetilde{O}(m \sqrt n)$，这是在不使用Lee-Sidford势垒或$\ell_1$内点法的情况下已知的最佳结果，展示了鲁棒内点法的惊人效力。作为推论，我们获得了一个$\widetilde{O}(\operatorname{tw}^3 \cdot m)$时间算法，用于在给定具有$m$条边的图时，计算宽度为$O(\operatorname{tw}\cdot \log(n))$的树分解。