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 2007), and unit-circle approximation for directed graphs (Ahmadinejad et. al. FOCS 2020). Moreover, SV approximation enjoys several useful properties not known to be possessed by previous notions of approximation, such as being preserved under products of random-walk matrices and with matrices of bounded norm. Notably, we show that there is a simple black-box reduction from SV-sparsifying Eulerian directed graphs to SV-sparsifying undirected graphs. With this reduction in hand, we provide a nearly linear-time algorithm for SV-sparsifying undirected and hence also Eulerian directed graphs. This also yields the first nearly linear-time algorithm for unit-circle-sparsifying Eulerian directed graphs. In addition, we give a nearly linear-time algorithm for SV-sparsifying (and UC-sparsifying) random-walk polynomials of Eulerian directed graphs with second normalized singular value bounded away from $1$ by $1/\text{poly}(n)$. Finally, we show that a simple repeated-squaring and sparsification algorithm for solving Laplacian systems, introduced by (Peng Spielman STOC 2014) for undirected graphs, also works for Eulerian digraphs whose random-walk matrix is normal (i.e. unitarily diagonalizable), if we use SV-sparsification at each step. Prior Laplacian solvers for Eulerian digraphs are significantly more complicated.
翻译:在本文中,我们引入了方向图之间的新的光谱近似概念, 我们称之为 Singal 值( SV) 近似。 SV 的近似比文献中考虑的先前光近近近概念更强, 包括无方向图( Spielman Teng STOC 2004) 的拉普拉西亚光谱近近近近, 定向图的标准近近近近( Cohen 等人 STOC 2007), 定向图( Ahmadinejad等人 2020 ) 的单位- 环球近似近似。 此外, SV 近似具有以前近似近似近似近似近似近似近似近似近似近似于近似于近似于于随机矩阵矩阵的光近光近光近光光光近光近光光光近光近光近光近光近光度的 Sliver 平面图( 以我们直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径算算算) 的S- 直径直径直径直径直径直路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路