A Nearly-Linear Time Algorithm for Linear Programs with Small Treewidth: A Multiscale Representation of Robust Central Path

Arising from structural graph theory, treewidth has become a focus of study in fixed-parameter tractable algorithms in various communities including combinatorics, integer-linear programming, and numerical analysis. Many NP-hard problems are known to be solvable in $\widetilde{O}(n \cdot 2^{{O(\mathrm{tw})})$} time, where $\mathrm{tw}$ is the treewidth of the input graph. Analogously, many problems in P should be solvable in $\widetilde{O}(n \cdot \mathrm{tw}^{O(1)})$ time; however, due to the lack of appropriate tools, only a few such results are currently known. [Fom+18] conjectured this to hold as broadly as all linear programs; in our paper, we show this is true: Given a linear program of the form $\min{Ax=b,\ell \leq x\leq u} c^{\top} x$, and a width-$\tau$ tree decomposition of a graph $GA$ related to $A$, we show how to solve it in time $$\widetilde{O}(n \cdot \tau² \log (1/\varepsilon)),$$ where $n$ is the number of variables and $\varepsilon$ is the relative accuracy. Combined with recent techniques in vertex-capacitated flow [BGS21], this leads to an algorithm with $\widetilde{O}(n^{1+o(1)} \cdot \mathrm{tw}² \log (1/\varepsilon))$ run-time. Besides being the first of its kind, our algorithm has run-time nearly matching the fastest run-time for solving the sub-problem $Ax=b$ (under the assumption that no fast matrix multiplication is used). We obtain these results by combining recent techniques in interior-point methods (IPMs), sketching, and a novel representation of the solution under a multiscale basis similar to the wavelet basis.