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+ε)-sparsifiers with
O(nε−2) edges exist [Batson-Spielman-Srivastava, STOC '09]
(which is optimal), the best known constructions of Eulerian sparsifiers
require Ω(nε−2log4n) 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 G, there exist an
Eulerian sparsifier with O(nε−2log2nlog2logn+nε−4/3log8/3n) 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 m1+δ-time algorithm for any constant
δ>0 for constructing Eulerian sparsifiers with
O(nε−2log3n) edges