# Singular Value Approximation and Reducing Directed to Undirected Graph Sparsification

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/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.

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