Twice-Ramanujan Sparsifiers [article]

Joshua Batson, Daniel A. Spielman, Nikhil Srivastava
2009 arXiv   pre-print
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 ∈^V, x^TL_Gx ≤ x^TL_Hx ≤
more » ... +1-2√(d))· x^TL_Gx 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.
arXiv:0808.0163v3 fatcat:szswkz4lcrccjmgeoc2lr7vbbi