Twice-Ramanujan Sparsifiers
Joshua Batson, Daniel A. Spielman, Nikhil Srivastava
TL;DR
The paper tackles spectral graph sparsification by proving that every graph on $n$ vertices admits a weighted sparsifier $H$ with $O(n)$ edges such that $x^T L_G x \le x^T L_H x \le \kappa x^T L_G x$ for all $x$, where $\kappa = \frac{d+1+2\sqrt{d}}{d+1-2\sqrt{d}}$. It introduces a deterministic, barrier-function approach to select a small subset of rank-one updates that preserve conditioning, yielding a polynomial-time algorithm to construct $H$ with at most $\lceil d(n-1)\rceil$ edges. The method generalizes Ramanujan expanders to arbitrary graphs and, in the complete-graph case, yields irregular, weighted expanders with strong spectral guarantees. The work also connects to Kadison–Singer and paving results, highlighting deep ties between spectral sparsification and foundational operator theory. The practical impact is a deterministic, near-optimal, linear-edge sparsifier that can underpin faster solvers for linear systems and scalable graph algorithms.
Abstract
We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. As linear-sized spectral sparsifiers of complete graphs are expanders, our sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs. In particular, we prove that for every $d>1$ and every undirected, weighted graph $G=(V,E,w)$ on $n$ vertices, there exists a weighted graph $H=(V,F,\tilde{w})$ with at most $\lceil d(n-1) \rceil$ edges such that for every $x \in \mathbb{R}^{V}$, \[ x^{T}L_{G}x \leq x^{T}L_{H}x \leq (\frac{d+1+2\sqrt{d}}{d+1-2\sqrt{d}})\cdot x^{T}L_{G}x \] where $L_{G}$ and $L_{H}$ are the Laplacian matrices of $G$ and $H$, respectively. Thus, $H$ approximates $G$ spectrally at least as well as a Ramanujan expander with $dn/2$ edges approximates the complete graph. We give an elementary deterministic polynomial time algorithm for constructing $H$.