Faster Spectral Sparsification and Numerical Algorithms for SDD Matrices

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 G˜ that approximates G in an algebraic sense. Concretely, for all vectors x and any ϵ > 0, the graph G˜ satisfies (1-ϵ ) x T L G x ≤ x T L G˜ x ≤ (1+ϵ) x T L G x , where L G and G˜ are the Laplacians of G and G˜ respectively. The first contribution of this article applies to all existing sparsification algorithms that rely on solving solving linear systems on graph Laplacians. These algorithms are the fastest known to date. Specifically, we show that less precision is required in the solution of the linear systems, leading to speedups by an O (log n ) factor. We also present faster sparsification algorithms for slightly dense graphs: — An O ( m log n ) time algorithm that generates a sparsifier with O ( n log 3 n /ϵ 2 ) edges. — An O ( m log log n ) time algorithm for graphs with more than n log 5 n log log n edges. — An O ( m ) algorithm for graphs with more than n log 10 n edges. — An O ( m ) algorithm for unweighted graphs with more than n log 8 n edges. These bounds hold up to factors that are in O ( poly (log log n )) and are conjectured to be removable.