Twice-Ramanujan Sparsifiers
Proceedings of the 41st annual ACM symposium on Theory of computing, Seite 255--262. New York, NY, USA, ACM, (2009)DOI: 10.1145/1536414.1536451
Zusammenfassung
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 ∈ RV, xT LG x ≤ xT LH x ≤ ((d+1+2√d)/(d+1-2√d)) • xT LG x 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.
