Solving Directed Laplacian Systems in Nearly-Linear Time through Sparse LU Factorizations

In this paper, we show how to solve directed Laplacian systems in nearly-linear time. Given a linear system in an n × n Eulerian directed Laplacian with m nonzero entries, we show how to compute an ε-approximate solution in time O(m log ^O(1) (n) log (1/ε)). Through reductions from [Cohen et al. FOCS'16], this gives the first nearly-linear time algorithms for computing ε-approximate solutions to row or column diagonally dominant linear systems (including arbitrary directed Laplacians) and computing ε-approximations to various properties of random walks on directed graphs, including stationary distributions, personalized PageRank vectors, hitting times, and escape probabilities. These bounds improve upon the recent almost-linear algorithms of [Cohen et al. STOC'17], which gave an algorithm to solve Eulerian Laplacian systems in time O((m+n2 ^{O(√ log n log log n)} )log ^O(1) (n ε ^-1 )). To achieve our results, we provide a structural result that we believe is of independent interest. We show that Eulerian Laplacians (and therefore the Laplacians of all strongly connected directed graphs) have sparse approximate LU-factorizations. That is, for every such directed Laplacian there are lower upper triangular matrices each with at most Õ(n) nonzero entries such that there product spectrally approximates the directed Laplacian in an appropriate norm. This claim can be viewed as an analog of recent work on sparse Cholesky factorizations of Laplacians of undirected graphs. We show how to construct such factorizations in nearly-linear time and prove that once constructed they yield nearly-linear time algorithms for solving directed Laplacian systems.