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 $\mathcal{T}_{\rm S}(m)$ to output a sparsifier with $\mathcal{N}_{\rm S}(n)$ edges, our algorithm solves a directed Eulerian system on $n$ vertices and $m$ edges to $\epsilon$ relative accuracy in time $$ O(\mathcal{T}_{\rm S}(m) + {\mathcal{N}_{\rm S}(n)\log {n}\log(n/\epsilon)}) + \tilde{O}(\mathcal{T}_{\rm S}(\mathcal{N}_{\rm S}(n)) \log n), $$ where the $\tilde{O}(\cdot)$ notation hides $\log\log(n)$ 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 that runs in $O(n \log^{5}n \log(n / \epsilon))$ time after $O(n^2 \log^{O(1)} n)$ pre-processing. At the core of our analyses are constructions of augmented matrices whose Schur complements encode error matrices.

翻译：我们证明，来自[Kyng-Lee-Peng-Sachdeva-Spielman STOC'16]的用于求解无向拉普拉斯线性系统的稀疏块消去算法可直接适用于有向拉普拉斯矩阵。给定一个稀疏化算法，该算法在具有$n$个顶点和$m$条边的图上耗时$\mathcal{T}_{\rm S}(m)$，输出包含$\mathcal{N}_{\rm S}(n)$条边的稀疏化图，我们的算法能在时间 $$ O(\mathcal{T}_{\rm S}(m) + {\mathcal{N}_{\rm S}(n)\log {n}\log(n/\epsilon)}) + \tilde{O}(\mathcal{T}_{\rm S}(\mathcal{N}_{\rm S}(n)) \log n) $$ 内求解一个$n$顶点$m$边上的有向欧拉系统，达到$\epsilon$相对精度，其中$\tilde{O}(\cdot)$记号隐藏了$\log\log(n)$因子。结合先前结果，这改进了强连通有向图、PageRank矩阵和非对称M矩阵中线性系统的求解运行时间。当与基于短环分解的较小欧拉稀疏化图的较慢构造方法相结合时，该算法还可在$O(n^2 \log^{O(1)} n)$预处理后，以$O(n \log^{5}n \log(n / \epsilon))$时间运行求解器。我们分析的核心是构造增广矩阵，其Schur补编码了误差矩阵。