We present a $\widetilde{O}(m\sqrt{\tau}+n\tau)$ time algorithm for finding a minimum-cost flow in graphs with $n$ vertices and $m$ edges, given a tree decomposition of width $\tau$ and polynomially bounded integer costs and capacities. This improves upon the current best algorithms for general linear programs bounded by treewidth which run in $\widetilde{O}(m \tau^{(\omega+1)/2})$ time by [Dong-Lee-Ye,21] and [Gu-Song,22], 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. As a corollary, for any graph $G$ with $n$ vertices, $m$ edges, and treewidth $\tau$, we obtain a $\widetilde{O}(\tau^3 \cdot m)$ time algorithm to compute a tree decomposition of $G$ with width $O(\tau \cdot \log n)$.

翻译：我们提出了一种时间复杂度为$\widetilde{O}(m\sqrt{\tau}+n\tau)$的算法，用于在具有$n$个顶点和$m$条边的图中找到最小费用流，其中给定宽度为$\tau$的树分解以及多项式有界的整数费用和容量。这改进了当前基于树宽的通用线性规划最优算法，即[Dong-Lee-Ye,21]和[Gu-Song,22]中运行时间为$\widetilde{O}(m \tau^{(\omega+1)/2})$的算法，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的最新进展。作为推论，对于任意具有$n$个顶点、$m$条边和树宽$\tau$的图$G$，我们获得了一个时间复杂度为$\widetilde{O}(\tau^3 \cdot m)$的算法，用于计算$G$的宽度为$O(\tau \cdot \log n)$的树分解。