Twice-Ramanujan Sparsifiers

Опубликовано 18 августа 2016, 22:57

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,\tilde{w}) with at most dn edges such that for every x \in \R^{V}, x^T L_G x \leq x^T L_H x \leq ( d+1+2\sqrt{d} / d+1-2\sqrt{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. Joint work with Josh Batson and Dan Spielman.