arXiv:2611.14342 · v1

Directed Singular-Value Sparsifiers with $O(n \log n/ε^2)$ Arcs

具有 O(n log n/ε²) 条弧的有向奇异值稀疏化器

首次提交 2026-11-11 01:08

版本更新 2026-11-11 01:08

雷达观测 2026-11-11 13:07

摘要

Given an unweighted directed graph, a sparsifier is a nonnegative reweighting of original arcs that preserves every in-degree and out-degree exactly. In this work, we study singular-value sparsifiers, which are generalizations of spectral sparsifiers. Singular-value sparsifiers find a number of applications in theoretical computer science, such as maintaining directed cuts and flows, speeding up directed Laplacian solvers, and sparsifying dense weight matrices in neural networks. Strong singular-value (SV) approximation supports sparse representations of directed random walks with guarantees that remain stable under products, and controls bilinear error in the left and right Schur-complement metrics. We prove that every unweighted directed graph on $n$ vertices admits an $\varepsilon$-SV sparsifier with $O(n\log(2n)/\varepsilon^2)$ arcs, improving the $O(n\log^5 n/\varepsilon^2)$ support bound of Lau, Wang, and Zhou. The sparsifier is a nonnegative reweighting of original arcs and preserves every in-degree and out-degree exactly. We also give a randomized construction with $\tilde{O}(m+n)$ real-arithmetic running time, where $m$ is the input support size, and the output has the same support bound and rational weights of polynomial bit length.

作者备注：44 pages