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 O(nβ 2O(tw)) time, where tw is the treewidth of the input
graph. Analogously, many problems in P should be solvable in O(nβ 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,ββ€xβ€uβcβ€x, and a width-Ο tree decomposition of a graph GAβ related to A, we
show how to solve it in time O(nβ Ο2log(1/Ξ΅)), where n is the number of variables and Ξ΅ is
the relative accuracy. Combined with recent techniques in vertex-capacitated
flow [BGS21], this leads to an algorithm with O(nβ tw2log(1/Ξ΅)) 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