GRAPH SPARSIFICATION, SPECTRAL SKETCHES, AND FASTER RESISTANCE COMPUTATION VIA SHORT CYCLE DECOMPOSITIONS\ast

We develop a framework for graph sparsification and sketching, based on a new tool, short cycle decomposition, which is a decomposition of an unweighted graph into an edgedisjoint collection of short cycles, plus a small number of extra edges. A simple observation shows that every graph G on n vertices with m edges can be decomposed in O(mn) time into cycles of length at most 2 log n, and at most 2n extra edges. We give an (m(1+o(1)))- time algorithm for constructing a short cycle decomposition, with cycles of length n(o(perpendicular to)), and n(1+o(perpendicular to)) extra edges. Both the existential and algorithmic variants of this decomposition enable us to make the following progress on several open problems in randomized graph algorithms: (1) We present an algorithm that runs in time m(1+o(1))epsilon(-1.5) and returns (1 +/- epsilon)-approximations to effective resistances of all edges, improving over the previous best runtime of O(min{m epsilon(-2),n(2) epsilon(-1)). This routine in turn gives an algorithm for approximating the determinant of a graph Laplacian up to a factor of (1 \pm \varepsilon) in m1+o(1) + n15/8+o(1)\varepsilon 7/4 time. (2) We show the existence of graphical spectral sketches with about n\varepsilon 1 edges, and also give efficient algorithms to construct them. A graphical spectral sketch is a distribution over sparse graphs H such that for a fixed vector \bfitx, we have \bfitx \top \bfitL H\bfitx = (1\pm \varepsilon)\bfitx \top \bfitL G\bfitx and \bfitx \top \bfitL + H \bfitx = (1 \pm \varepsilon)\bfitx \top \bfitL + G \bfitx with high probability, where \bfitL is the graph Laplacian and \bfitL + is its pseudoinverse. This implies the existence of resistance sparsifiers with about n\varepsilon 1 edges that preserve the effective resistance between every pair of vertices up to (1\pm \varepsilon). (3) By combining short cycle decompositions with known tools in graph sparsification, we show the existence of nearly linear sized degree-preserving spectral sparsifiers, as well as significantly sparser approximations of Eulerian directed graphs. The latter is critical to recent breakthroughs on faster algorithms for solving linear systems in directed Laplacians. The running time and output qualities of our spectral sketch and degree-preserving (directed) sparsification algorithms are limited by the efficiency of our routines for constructing short cycle decompositions. Improved algorithms for short cycle decompositions will lead to improvement in each of these algorithms.