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 $\tilde{G}$ 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.