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 widetildeO(n · 2O(tw)) time, where tw is the treewidth of the input graph. Analogously, many problems in P should be solvable in widetildeO(n · twO(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 minAx=b,ℓ ≤ x≤ u ctop x, and a width-τ tree decomposition of a graph GA related to A, we show how to solve it in time widetildeO(n · τ2 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 widetildeO(n1+o(1) · tw2 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.