3 Barrier Functions

In the previous lectures, we explored the problem of finding spectral sparsifiers and saw both deterministic and randomized algorithms that given a graph G with n vertices finds a spectral sparsifier with O ( n logn 2 ) edges. In this lecture, we improve upon this result to find linear-sized spectral sparsifiers, or more concretely, spectral sparsifiers with O ( n 2 ) edges. This is an interesting result since we noted in the past that a spectral sparsifier is a generalization of the cut sparsifier of a graph, and the best results known previous to this one for cut sparsifiers were that a cut sparsifier with n log n/ 2 edges can be found in nearly linear time [1]. Recall that given a graph G = (V,E), a weighted graph H = (V,E ′) with weights w(i, j) is an -spectral sparsifier of G if