In numerical linear algebra, considerable effort has been devoted to obtaining faster algorithms for linear systems whose underlying matrices exhibit structural properties. A prominent success story is the method of generalized nested dissection~[Lipton-Rose-Tarjan'79] for separable matrices. On the other hand, the majority of recent developments in the design of efficient linear program (LP) solves do not leverage the ideas underlying these faster linear system solvers nor consider the separable structure of the constraint matrix. We give a faster algorithm for separable linear programs. Specifically, we consider LPs of the form $\min_{\mathbf{A}\mathbf{x}=\mathbf{b}, \mathbf{l}\leq\mathbf{x}\leq\mathbf{u}} \mathbf{c}^\top\mathbf{x}$, where the graphical support of the constraint matrix $\mathbf{A} \in \mathbb{R}^{n\times m}$ is $O(n^\alpha)$-separable. These include flow problems on planar graphs and low treewidth matrices among others. We present an $\tilde{O}((m+m^{1/2 + 2\alpha}) \log(1/\epsilon))$ time algorithm for these LPs, where $\epsilon$ is the relative accuracy of the solution. Our new solver has two important implications: for the $k$-multicommodity flow problem on planar graphs, we obtain an algorithm running in $\tilde{O}(k^{5/2} m^{3/2})$ time in the high accuracy regime; and when the support of $\mathbf{A}$ is $O(n^\alpha)$-separable with $\alpha \leq 1/4$, our algorithm runs in $\tilde{O}(m)$ time, which is nearly optimal. The latter significantly improves upon the natural approach of combining interior point methods and nested dissection, whose time complexity is lower bounded by $\Omega(\sqrt{m}(m+m^{\alpha\omega}))=\Omega(m^{3/2})$, where $\omega$ is the matrix multiplication constant. Lastly, in the setting of low-treewidth LPs, we recover the results of [DLY,STOC21] and [GS,22] with significantly simpler data structure machinery.

翻译：在数值线性代数中，大量研究致力于为具有结构特性的矩阵的线性系统设计更快速的算法。一个突出的成功案例是用于可分离矩阵的广义嵌套剖分方法[Lipton-Rose-Tarjan'79]。然而，近期大多数高效线性规划求解算法的设计并未借鉴这些快速线性系统求解器的核心思想，也未考虑约束矩阵的可分离结构。本文提出了一种针对可分离线性规划的更快速算法。具体而言，我们考虑形如 $\min_{\mathbf{A}\mathbf{x}=\mathbf{b}, \mathbf{l}\leq\mathbf{x}\leq\mathbf{u}} \mathbf{c}^\top\mathbf{x}$ 的线性规划，其中约束矩阵 $\mathbf{A} \in \mathbb{R}^{n\times m}$ 的图形支撑是 $O(n^\alpha)$-可分离的。这类问题包括平面图上的流问题以及低树宽矩阵等情形。我们给出了一个运行时间为 $\tilde{O}((m+m^{1/2 + 2\alpha}) \log(1/\epsilon))$ 的算法，其中 $\epsilon$ 是解的相对精度。我们的新求解器有两个重要应用：对于平面图上的 $k$-多商品流问题，在高精度场景下我们得到运行时间为 $\tilde{O}(k^{5/2} m^{3/2})$ 的算法；当 $\mathbf{A}$ 的支撑为 $O(n^\alpha)$-可分离且 $\alpha \leq 1/4$ 时，我们的算法运行时间为 $\tilde{O}(m)$，这几乎是渐近最优的。后者显著改进了结合内点法和嵌套剖分的自然方法——该方法的复杂度下界为 $\Omega(\sqrt{m}(m+m^{\alpha\omega}))=\Omega(m^{3/2})$，其中 $\omega$ 是矩阵乘法常数。最后，在低树宽线性规划场景下，我们以显著简化的数据结构机制复现了[DLY, STOC21]和[GS, 22]的结果。