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

TL;DR

This paper introduces a new short cycle decomposition framework for graphs, enabling faster algorithms for resistance approximation, spectral sketching, and graph sparsification, with applications to directed Laplacian systems.

Contribution

It presents a novel short cycle decomposition method and applies it to improve algorithms for resistance estimation, spectral sketching, and sparsification.

Findings

Faster resistance approximation algorithm with $m^{1+o(1)} ext{ time}$

Existence of spectral sketches preserving quadratic forms

Construction of nearly-linear size degree-preserving spectral sparsifiers

Abstract

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 few 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. These decompositions enable us to make progress on several open questions: * We give an algorithm to find $(1\pm\epsilon)$-approximations to effective resistances of all edges in time $m^{1+o(1)}\epsilon^{-1.5}$, improving over the previous best of $\tilde{O}(\min\{m\epsilon^{-2},n^2 \epsilon^{-1}\})$. This gives an algorithm to approximate the determinant of a Laplacian up to $(1\pm\epsilon)$ in $m^{1 + o(1)} + n^{15/8+o(1)}\epsilon^{-7/4}$ time. * We show existence and efficient algorithms for constructing graphical spectral sketches -- a distribution over sparse graphs H such that for a fixed vector $x$, we have w.h.p. $x'L_Hx=(1\pm\epsilon)x'L_Gx$ and $x'L_H^+x=(1\pm\epsilon)x'L_G^+x$. 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 $(1\pm\epsilon).$ * 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 directed graphs. The latter is critical to recent breakthroughs on faster algorithms for solving linear systems in directed Laplacians. Improved algorithms for constructing short cycle decompositions will lead to improvements for each of the above results.