Faster spectral sparsification and numerical algorithms for SDD matrices

TL;DR

This paper introduces faster algorithms for spectral graph sparsification, significantly reducing computation time for creating sparse approximations of graphs, which enhances the efficiency of solving linear systems and eigenvector computations for SDD matrices.

Contribution

The paper presents the fastest known algorithms for spectral graph sparsification, achieving improved running times for various graph densities and edge counts, and applies these to accelerate numerical algorithms for SDD matrices.

Findings

Faster sparsification algorithms with $ ilde{O}(m ext{ or }m ext{ log } n)$ time.

Sparsifiers with $O(n ext{ log } n/ ext{epsilon}^2)$ edges computed more efficiently.

Accelerated algorithms for solving linear systems and eigenvector computations in SDD matrices.

Abstract

We study algorithms for spectral graph sparsification. The input is a graph $G$ with $n$ vertices and $m$ edges, and the output is a sparse graph $\tilde{G}$ that approximates $G$ in an algebraic sense. Concretely, for all vectors $x$ and any $\epsilon>0$, $\tilde{G}$ satisfies $$ (1-\epsilon) x^T L_G x \leq x^T L_{\tilde{G}} x \leq (1+\epsilon) x^T L_G x, $$ where $L_G$ and $L_{\tilde{G}}$ are the Laplacians of $G$ and $\tilde{G}$ respectively. We show that the fastest known algorithm for computing a sparsifier with $O(n\log n/\epsilon^2)$ edges can actually run in $\tilde{O}(m\log^2 n)$ time, an $O(\log n)$ factor faster than before. We also present faster sparsification algorithms for slightly dense graphs. Specifically, we give an algorithm that runs in $\tilde{O}(m\log n)$ time and generates a sparsifier with $\tilde{O}(n\log^3{n}/\epsilon^2)$ edges. This implies that a sparsifier with $O(n\log n/\epsilon^2)$ edges can be computed in $\tilde{O}(m\log n)$ time for graphs with more than $O(n\log^4 n)$ edges. We also give an $\tilde{O}(m)$ time algorithm for graphs with more than $n\log^5 n (\log \log n)^3$ edges of polynomially bounded weights, and an $O(m)$ algorithm for unweighted graphs with more than $n\log^8 n (\log \log n)^3 $ edges and $n\log^{10} n (\log \log n)^5$ edges in the weighted case. The improved sparsification algorithms are employed to accelerate linear system solvers and algorithms for computing fundamental eigenvectors of slightly dense SDD matrices.