Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

Abstract

We present the first almost-linear time algorithm for constructing linear-sized spectral sparsification for graphs. This improves all previous constructions of linear-sized spectral sparsification, which requires $\hat{I}\text{c◯}(n^2)$ time. A key ingredient in our algorithm is a novel combination of two techniques used in literature for constructing spectral sparsification: Random sampling by effective resistance, and adaptive constructions based on barrier functions.

22 pages. A preliminary version of this paper is to appear in proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015)