Twice-Ramanujan Sparsifiers

Abstract

A sparsifier of a graph is a sparse graph that approximates it. A spectral sparsifier is one that approximates it spectrally, which means that their Laplacian matrices have similar quadratic forms. We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. In particular, we prove that for every $\epsilon \in (0,1)$ and every undirected, weighted graph $G = (V,E,w)$ on $n$ vertices, there exists a weighted graph $H=(V,F,\tilde{w})$ with at most $\lceil (n-1)/\epsilon^2\rceil$ edges such that for every $x \in \R^{V}$, $ (1-\epsilon)^2 \cdot x^T L_G x \leq x^{T} L_{H} x \leq (1+\epsilon)^2 \cdot x^{T} L_{G} x$, where $L_{G}$ and $L_{H}$ are the Laplacian matrices of $G$ and $H$, respectively. We give an elementary deterministic polynomial time algorithm for constructing $H$. This result is a special case of a significantly more general theorem which provides sparse approximations of general positive semidefinite matrices: given any real matrix $B_{n\times m}$...