A nearly-linear time algorithm for linear programs with small treewidth: a multiscale representation of robust central path

ABSTRACTArising from structural graph theory, treewidth has become a focus of study in fixed-parameter tractable algorithms. Many NP-hard problems are known to be solvable in O(n · 2O(τ)) time, where τ is the treewidth of the input graph. Analogously, many problems in P should be solvable in O(n · τO(1)) time; however, due to the lack of appropriate tools, only a few such results are currently known. In our paper, we show this holds for linear programs: Given a linear program of the form minAx=b,ℓ ≤ x≤ u c⊤ x whose dual graph GA has treewidth τ, and a corresponding width-τ tree decomposition, we show how to solve it in time O(n · τ2 log(1/ε)), where n is the number of variables and ε is the relative accuracy. When a tree decomposition is not given, we use existing techniques in vertex separators to obtain algorithms with O(n · τ4 log(1/ε)) and O(n · τ2 log(1/ε) + n1.5) run-times. 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.