Sparsifying Cayley Graphs on Every Group

2025-08-11

AI Summary

This work addresses spectral sparsification of Cayley graphs over arbitrary groups $G$: does there exist a sparse graph—a spectral $varepsilon$-sparsifier—with only $O(mathrm{polylog}|G|/varepsilon^2)$ reweighted generator edges that spectrally approximates the original Cayley graph? The authors establish, for the first time, that *all* Cayley graphs—including those over non-Abelian groups—admit near-optimal spectral sparsifiers, and provide an efficient construction. They further extend the result to directed Cayley graphs, achieving cut sparsification. Technically, the approach integrates spectral graph theory, representation theory of finite groups, and randomized reweighting techniques. A key conceptual breakthrough is the proof that linear systems arising from non-Abelian group algebras cannot be approximated by polynomial-size graph sparsifiers—thereby establishing a fundamental theoretical separation between graph sparsification and linear system sparsification, and highlighting the distinctive structural sparsifiability inherent to Cayley graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Existence of Cayley graph sparsifiers preserving polylog edges
• Efficient algorithm for spectral sparsifiers of Cayley graphs
• Separation between graph sparsification and linear equation sparsification

Innovation

• Constructs Cayley graph spectral sparsifiers efficiently
• Extends algorithm to directed Cayley graphs
• Demonstrates non-abelian linear equations sparsification limits