有向图上的奇异值近似与稀疏化随机游走
Singular Value Approximation and Sparsifying Random Walks on Directed Graphs
AI总结：
本文提出有向图的奇异值（SV）近似新概念，给出欧拉有向图及其随机游走的近线性时间稀疏化算法，并实现任意有向图割平稳概率质量的近线性时间近似。
摘要：
在本文中，我们引入了一种新的、基于谱的有向图之间近似概念，称之为奇异值（SV）近似。SV近似比文献中先前考虑的谱近似概念更强，包括无向图拉普拉斯算子的谱近似（Spielman Teng STOC 2004）、有向图的标准近似（Cohen et. al. STOC 2017）以及有向图的单位圆近似（Ahmadinejad et. al. FOCS 2020）。此外，SV近似具有先前近似概念所不具备的若干有用性质，例如，它在随机游走矩阵和有界矩阵的乘积下保持不变。我们提供了一个近线性时间算法，用于对欧拉有向图以及此类图上的$\ell$步随机游走进行SV稀疏化（从而也实现UC稀疏化），其中任意$\ell\leq \text{poly}(n)$。结合（Cohen et. al. FOCS 2018）的欧拉缩放算法，给定任意（不一定欧拉的）有向图和一个顶点集合$S$，我们可以在近线性时间内近似$\ell$步随机游走中$(S,S^c)$割的平稳概率质量，使其乘性误差为$1/\text{polylog}(n)$，加性误差为$1/\text{poly}(n)$。作为这些结果的起点，我们提供了一个简单的黑盒归约，将欧拉有向图的SV稀疏化归约为无向图的SV稀疏化；对于先前的谱近似概念，这种有向到无向的归约尚不为人所知。
英文摘要：
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.
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