Optimal Sparsifiers for Abelian Cayley Graphs

2026-07-09

AI Summary

This work addresses the problem of spectral sparsification of Cayley graphs over finite abelian groups. By reformulating sparsification as a lower bound estimation on the volume of a convex body and leveraging the symmetry of group characters together with ℓ₁-sparsification techniques, the authors construct—for the first time—a weighted Cayley graph sparsifier using only O(log|G|) generators, and prove this bound is optimal for abelian groups. Notably, when G = 𝔽₂ⁿ, the resulting code sparsifier has size O(n/ε²), which improves upon prior constructions that incurred polylogarithmic redundancy factors, thereby significantly advancing the state of the art in sparsification for 𝔽₂-linear codes.

Abstract

We prove that for every Cayley graph $\mathcal{G}$ over any finite abelian group $G$, there is a weighted Cayley graph with $O(\log |G|)$ generators that is a spectral sparsifier for $\mathcal{G}$. This bound is optimal. Applying our bound to the group $G = \mathbb{F}_2^n$, yields, as a corollary, $O(n/\varepsilon^2)$-sized code sparsifiers for $\mathbb{F}_2$-linear codes, improving on the work of Khanna, Putterman and Sudan (SODA'24) who obtained a similar result with an additional $\mathrm{polylog}(n)$ loss. Our proof is strongly inspired by a recent work of Reis and Rothvoss for the construction of $\ell_1$-sparsifiers. Following their work, the abelian Cayley sparsification problem can be reduced to establishing a lower bound for the volume of a certain natural convex body. This volume bound follows from a short, elementary argument that relies on character symmetry.

Problem

• spectral sparsifier
• Cayley graph
• abelian group
• optimal sparsification
• finite group

Innovation

• spectral sparsification
• Abelian Cayley graphs
• convex body volume
• character symmetry
• linear codes