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 $\Omega(n\varepsilon^{-2}\log^4 n)$ 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 $m^{1+\delta}$-time algorithm for any constant $\delta > 0$ for constructing Eulerian sparsifiers with $O(n\varepsilon^{-2}\log^3 n)$ edges.

翻译：对于有向欧拉图进行谱稀疏化是设计快速算法求解有向拉普拉斯线性系统的关键组成部分。有向拉普拉斯线性系统求解器是实现随机游走中基本问题（如计算平稳分布、命中时间与通勤时间、个性化PageRank向量）快速计算的重要算法原语。虽然无向图的谱稀疏化问题已得到充分研究，且已知对任意图$G$存在边数为$O(n\varepsilon^{-2})$的$(1+\varepsilon)$-稀疏化方法[Batson-Spielman-Srivastava, STOC '09]（该结果已达最优），但当前已知最优的欧拉图稀疏化构造需要$\Omega(n\varepsilon^{-2}\log^4 n)$条边，且基于短环分解技术[Chu et al., FOCS '18]。本文对欧拉图稀疏化构造进行了改进，具体包括：1. 证明对任意有向欧拉图$\vec{G}$，存在边数为$O(n\varepsilon^{-2} \log^2 n \log^2\log n + n\varepsilon^{-4/3}\log^{8/3} n)$的欧拉稀疏化方法。该结果通过结合短环分解技术[Chu-Gao-Peng-Sachdeva-Sawlani-Wang, FOCS '18, SICOMP]与[Parter-Yogev, ICALP '19]，以及矩阵Spencer猜想的最新进展[Bansal-Meka-Jiang, STOC '23]实现。2. 对基于短环分解的构造方法给出改进分析，对任意常数$\delta > 0$，提出了$m^{1+\delta}$时间复杂度的算法，可构造边数为$O(n\varepsilon^{-2}\log^3 n)$的欧拉稀疏化方法。