We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. As linearsized 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,w) with at most d(n − 1) edges such that for every x ∈ R V , LGx 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.
In a beautiful piece of work, REF showed that an ǫ-spectral sparsifier with a linear number of O(n/ǫ 2 ) edges exist and can be computed in polynomial time.
Twice-ramanujan sparsifiers
STOC '09
Mathematics Computer Science