Graph Sparsification, Spectral Sketches, and Faster Resistance Computation, via Short Cycle Decompositions

We develop a framework for graph sparsification and sketching, based on a new tool, short cycle decomposition a decomposition of an unweighted graph into an edge-disjoint collection of short cycles, plus a small number of extra edges. A simple observation gives 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(1)), and n(1+o(1)) extra edges. Both the existential and algorithmic variants of this decomposition enable us to make 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 of (O) over tilde (min{m epsilon(-2), n(2) epsilon(-1)}). This routine in turn gives an algorithm to approximate the determinant of a graph Laplacian up to a factor of (1 +/-epsilon) in m(1+o(1)) + n(15/8+o(1)) epsilon(-7/4) time. 2) We show existence and efficient algorithms for constructing graphical spectral sketches - a distribution over graphs H with about n epsilon(-1) edges such that for a fixed vector x, we have x(inverted perpendicular) L-H x = (1 +/- epsilon)x(inverted perpendicular) L-G x and x(inverted perpendicular) L-H(+) x = (1 +/- epsilon) x(inverted perpendicular) L-G(+) x with high probability, where L is the graph Laplacian and L+ is its pseudoinverse. This implies the existence of resistance-sparsifiers with about n epsilon(-1) edges that preserve the effective resistances between every pair of vertices up to (1 +/- epsilon). 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 improvements for each of these algorithms.