In this paper, we introduce a new, spectral notion of approximation between directed graphs, which we call singular value (SV) approximation. SV-approximation is stronger than previous notions of spectral approximation considered in the literature, including spectral approximation of Laplacians for undirected graphs (Spielman Teng STOC 2004), standard approximation for directed graphs (Cohen et. al. STOC 2017), and unit-circle approximation for directed graphs (Ahmadinejad et. al. FOCS 2020). Further, SV approximation enjoys several useful properties not possessed by previous notions of approximation, e.g., it is preserved under products of random-walk matrices and bounded matrices. We provide a nearly linear-time algorithm for SV-sparsifying (and hence UC-sparsifying) Eulerian directed graphs, as well as $\ell$-step random walks on such graphs, for any $\ell\leq \text{poly}(n)$. Combined with the Eulerian scaling algorithms of (Cohen et. al. FOCS 2018), given an arbitrary (not necessarily Eulerian) directed graph and a set $S$ of vertices, we can approximate the stationary probability mass of the $(S,S^c)$ cut in an $\ell$-step random walk to within a multiplicative error of $1/\text{polylog}(n)$ and an additive error of $1/\text{poly}(n)$ in nearly linear time. As a starting point for these results, we provide a simple black-box reduction from SV-sparsifying Eulerian directed graphs to SV-sparsifying undirected graphs; such a directed-to-undirected reduction was not known for previous notions of spectral approximation.

翻译：本文提出一种新的有向图之间的谱逼近概念，称为奇异值（SV）逼近。SV逼近强于文献中先前考虑的谱逼近概念，包括无向图拉普拉斯矩阵的谱逼近（Spielman Teng STOC 2004）、有向图的标准逼近（Cohen等人 STOC 2017）以及有向图的单位圆逼近（Ahmadinejad等人 FOCS 2020）。此外，SV逼近具有先前逼近概念所不具备的若干有用性质，例如它在随机游走矩阵与有界矩阵的乘积下保持不变。我们为欧拉有向图及其任意$\ell\leq \text{poly}(n)$步随机游走提供了一种近线性时间的SV稀疏化（进而UC稀疏化）算法。结合（Cohen等人 FOCS 2018）的欧拉缩放算法，对于任意（不必为欧拉图）有向图及顶点集$S$\u3000，我们可以在近线性时间内，以$1/\text{polylog}(n)$的乘法误差和$1/\text{poly}(n)$的加法误差逼近$\ell$步随机游走中$(S,S^c)$切割的平稳概率质量。作为这些结果的出发点，我们提供了一个从欧拉有向图SV稀疏化到无向图SV稀疏化的简单黑箱归约；这种从有向到无向的归约在先前谱逼近概念中尚未已知。