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 dn 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.
Joint work with Josh Batson and Dan Spielman.