Twice-Ramanujan sparsifiers

Abstract. We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. As linear-sized spectral sparsifiers of complete graphs are expanders, our sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs. In particular, we prove that for every d> 1 and every undirected, weighted graph G = (V, E,w) on n vertices, there exists a weighted graph H = (V, F, w̃) with at most d(n − 1) edges such that for every x ∈ RV, xTLGx ≤ xTLHx ≤ d+1+2 d d+1−2√d) · xTLGx, where LG and LH are the Laplacian matrices of G and H, respectively. Thus, H approximates G spectrally at least as well as a Ramanujan expander with dn/2 edges approximates the complete graph. We give an elementary deterministic polynomial time algorithm for constructing H