Twice-ramanujan sparsifiers

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,~{w}) with at most ⌈d(n-1)⌉ edges such that for every x ∈ R V , [x T L G x ≤ x T L H x ≤ ((d+1+2√d)/(d+1-2√d)) • x T L G x] where L G and L H 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.

Authors

• Joshua D. Batson
• Daniel A. Spielman
• Nikhil Srivastava

Keywords

• expander graphs
• spectral graph theory