Recent years have seen extensive research on directed graph sparsification. In this work, we initiate the study of fast fully dynamic spectral and cut sparsification algorithms for directed graphs. We introduce a new notion of spectral sparsification called degree-balance preserving spectral approximation, which maintains the difference between the in-degree and out-degree of each vertex. The approximation error is measured with respect to the corresponding undirected Laplacian. This notion is equivalent to direct Eulerian spectral approximation when the input graph is Eulerian. Our algorithm achieves an amortized update time of O(ε^{-2} ⋅ polylog(n)) and produces a sparsifier of size O(ε^{-2} n ⋅ polylog(n)). Additionally, we present an algorithm that maintains a constant-factor approximation sparsifier of size O(n ⋅ polylog(n)) against an adaptive adversary for O(polylog(n))-partially symmetrized graphs, a notion introduced in [Kyng-Meierhans-Probst Gutenberg '22]. A β-partial symmetrization of a directed graph G is the union of G and β ⋅ G, where G is the corresponding undirected graph of G. This algorithm also achieves a polylogarithmic amortized update time. Moreover, we develop a fully dynamic algorithm for maintaining a cut sparsifier for β-balanced directed graphs, where the ratio between weighted incoming and outgoing edges of any cut is at most β. This algorithm explicitly maintains a cut sparsifier of size O(ε^{-2}β n ⋅ polylog(n)) in worst-case update time O(ε^{-2}β ⋅ polylog(n)).

We provide an algorithm that, given an n-vertex m-edge Eulerian graph with polynomially bounded weights, computes an (n log2 n · ∈-2)-edge ε-approximate Eulerian sparsifier with high probability in (m log3 n ) time (where (·) hides polyloglog(n ) factors). Due to a reduction from [Peng-Song, STOC ’22], this yields an (m log3 n + n log6 n )-time algorithm for solving n-vertex m-edge Eulerian Laplacian systems with polynomially-bounded weights with high probability, improving upon the previous state- of-the-art runtime of Ω(m log8 n + n log23 n ). We also give a polynomial-time algorithm that computes O (min(n log n · ε-2 + n log5/3 n log3/2 n · ε-2))-edge sparsifiers, improving the best such sparsity bound of O (n log2 n · ε-2 + n log8/3 n · ε-4/3) [Sachdeva-Thudi-Zhao, ICALP ’24].

Spectral sparsification for directed Eulerian graphs is a key component in the design of fast algorithms for solving directed Laplacian linear systems. Directed Laplacian linear system solvers are crucial algorithmic primitives to fast computation of fundamental problems on random walks, such as computing stationary distribution, hitting and commute time, and personalized PageRank vectors. While spectral sparsification is well understood for undirected graphs and it is known that for every graph $G,$ $(1+\varepsilon)$-sparsifiers with $O(n\varepsilon^{-2})$ edges exist [Batson-Spielman-Srivastava, STOC '09] (which is optimal), the best known constructions of Eulerian sparsifiers require $Ω(n\varepsilon^{-2}\log^4 n)$ edges and are based on short-cycle decompositions [Chu et al., FOCS '18]. In this paper, we give improved constructions of Eulerian sparsifiers, specifically: 1. We show that for every directed Eulerian graph $\vec{G},$ there exist an Eulerian sparsifier with $O(n\varepsilon^{-2} \log^2 n \log^2\log n + n\varepsilon^{-4/3}\log^{8/3} n)$ edges. This result is based on combining short-cycle decompositions [Chu-Gao-Peng-Sachdeva-Sawlani-Wang, FOCS '18, SICOMP] and [Parter-Yogev, ICALP '19], with recent progress on the matrix Spencer conjecture [Bansal-Meka-Jiang, STOC '23]. 2. We give an improved analysis of the constructions based on short-cycle decompositions, giving an $m^{1+δ}$-time algorithm for any constant $δ> 0$ for constructing Eulerian sparsifiers with $O(n\varepsilon^{-2}\log^3 n)$ edges.

A symmetric matrix is called a Laplacian if it has nonpositive off-diagonal entries and zero row sums. Since the seminal work of Spielman and Teng (2004) on solving Laplacian linear systems in nearly linear time, several algorithms have been designed for the task. Yet, the work of Kyng and Sachdeva (2016) remains the simplest and most practical sequential solver. They presented a solver purely based on random sampling and without graph-theoretic constructions such as low-stretch trees and sparsifiers.