Sparsifying Cayley Graphs on Every Group

TL;DR

This paper proves that Cayley graphs over any group can be sparsified to preserve spectral properties with only polylogarithmic generators, providing an efficient algorithm and contrasting with the complexity of sparsifying linear equations over non-abelian groups.

Contribution

It establishes the existence of spectral sparsifiers for Cayley graphs over any group with polylogarithmic size, and provides an efficient algorithm for their construction, extending to directed graphs.

Findings

Cayley graphs admit spectral sparsifiers with polylogarithmic generators.

Efficient algorithms for constructing Cayley graph sparsifiers are developed.

Sparsification of linear equations over non-abelian groups requires super-polynomially many equations.

Abstract

A classic result in graph theory, due to Batson, Spielman, and Srivastava (STOC 2009) shows that every graph admits a $(1 \pm \varepsilon)$ cut (or spectral) sparsifier which preserves only $O(n / \varepsilon^2)$ reweighted edges. However, when applying this result to \emph{Cayley graphs}, the resulting sparsifier is no longer necessarily a Cayley graph -- it can be an arbitrary subset of edges. Thus, a recent line of inquiry, and one which has only seen minor progress, asks: for any group $G$, do all Cayley graphs over the group $G$ admit sparsifiers which preserve only $\mathrm{polylog}(|G|)/\varepsilon^2$ many re-weighted generators? As our primary contribution, we answer this question in the affirmative, presenting a proof of the existence of such Cayley graph spectral sparsifiers, along with an efficient algorithm for finding them. Our algorithm even extends to \emph{directed} Cayley graphs, if we instead ask only for cut sparsification instead of spectral sparsification. We additionally study the sparsification of linear equations over non-abelian groups. In contrast to the abelian case, we show that for non-abelian valued equations, super-polynomially many linear equations must be preserved in order to approximately preserve the number of satisfied equations for any input. Together with our Cayley graph sparsification result, this provides a formal separation between Cayley graph sparsification and sparsifying linear equations.