A Nearly-Linear Time Algorithm for Linear Programs with Small Treewidth: A Multiscale Representation of Robust Central Path
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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(n · 2^O(tw)) time, where tw is the treewidth of the input graph. Analogously, many problems in P should be solvable in O(n ·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,ℓ≤ x≤ u c^⊤ x, and a width-τ tree decomposition of a graph G_A related to A, we show how to solve it in time O(n ·τ^2 log (1/ε)), where n is the number of variables and ε is the relative accuracy. Combined with existing techniques in vertex separators, this leads to algorithms with O(n ·tw^4 log (1/ε)) and O(n ·tw^2 log (1/ε) + n^1.5) run-times when a tree decomposition is not given. 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. This representation further yields the first IPM with o(rank(A)) time per iteration when the treewidth is small.
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