Sparsified block elimination for directed laplacians

We show that the sparsified block elimination algorithm for solving undirected Laplacian linear systems from [Kyng-Lee-Peng-Sachdeva-Spielman STOC'16] directly works for directed Laplacians. Given access to a sparsification algorithm that, on graphs with n vertices and m edges, takes time TS(m) to output a sparsifier with NS(n) edges, our algorithm solves a directed Eulerian system on n vertices and m edges to є relative accuracy in time O(TS(m) + NS(n)lognlog(n/є)) + Õ(TS(NS(n)) logn), where the Õ(·) notation hides loglog(n) factors. By previous results, this implies improved runtimes for linear systems in strongly connected directed graphs, PageRank matrices, and asymmetric M-matrices. When combined with slower constructions of smaller Eulerian sparsifiers based on short cycle decompositions, it also gives a solver algorithm that, after pre-processing the matrix in O(n2 logO(1) n) time, takes O(n log5n log(n / є)) time per solve. At the core of our analyses are constructions of augmented matrices whose Schur complements encode error matrices.