Fast Algorithms for Separable Linear Programs

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.