A Sketching Algorithm for Spectral Graph Sparsification

TL;DR

This paper introduces a novel sketching algorithm for spectral graph sparsification that significantly reduces space complexity for approximating quadratic forms of graph Laplacians, especially for fixed vectors, surpassing previous bounds.

Contribution

The paper presents the first sub-quadratic space sketch for spectral sparsification that approximates quadratic forms for fixed vectors, improving upon prior linear-space methods.

Findings

Achieves O(n psilon^{-1.6}) bits for fixed vector approximation

Contrasts with lower bounds for general PSD matrices

Builds on recent work for cut query sketches

Abstract

We study the problem of compressing a weighted graph $G$ on $n$ vertices, building a "sketch" $H$ of $G$, so that given any vector $x \in \mathbb{R}^n$, the value $x^T L_G x$ can be approximated up to a multiplicative $1+\epsilon$ factor from only $H$ and $x$, where $L_G$ denotes the Laplacian of $G$. One solution to this problem is to build a spectral sparsifier $H$ of $G$, which, using the result of Batson, Spielman, and Srivastava, consists of $O(n \epsilon^{-2})$ reweighted edges of $G$ and has the property that simultaneously for all $x \in \mathbb{R}^n$, $x^T L_H x = (1 \pm \epsilon) x^T L_G x$. The $O(n \epsilon^{-2})$ bound is optimal for spectral sparsifiers. We show that if one is interested in only preserving the value of $x^T L_G x$ for a {\it fixed} $x \in \mathbb{R}^n$ (specified at query time) with high probability, then there is a sketch $H$ using only $\tilde{O}(n \epsilon^{-1.6})$ bits of space. This is the first data structure achieving a sub-quadratic dependence on $\epsilon$. Our work builds upon recent work of Andoni, Krauthgamer, and Woodruff who showed that $\tilde{O}(n \epsilon^{-1})$ bits of space is possible for preserving a fixed {\it cut query} (i.e., $x\in \{0,1\}^n$) with high probability; here we show that even for a general query vector $x \in \mathbb{R}^n$, a sub-quadratic dependence on $\epsilon$ is possible. Our result for Laplacians is in sharp contrast to sketches for general $n \times n$ positive semidefinite matrices $A$ with $O(\log n)$ bit entries, for which even to preserve the value of $x^T A x$ for a fixed $x \in \mathbb{R}^n$ (specified at query time) up to a $1+\epsilon$ factor with constant probability, we show an $\Omega(n \epsilon^{-2})$ lower bound.