Optimal Sparsifiers for Abelian Cayley Graphs

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 $G = \mathbb{F}_2<sup>n$, yields, as a corollary, $O(n/\varepsilon<sup>2)$-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.

• The paper establishes that O(log |G|/ε²) generators suffice for spectral sparsification with a tight bound that cannot be improved.
• It utilizes a novel volumetric analysis of a convex 'sparsification polytope' and leverages abelian group symmetry to iteratively reduce generators.
• The approach offers practical benefits for algorithm design and coding theory by generalizing expander constructions and preserving algebraic structure.

Optimal Spectral Sparsification for Abelian Cayley Graphs

Introduction and Context

The paper "Optimal Sparsifiers for Abelian Cayley Graphs" (2607.08261) establishes optimal spectral sparsification bounds for Cayley graphs over finite abelian groups. The result addresses a longstanding open question regarding the best possible generator set size required to sparsify any abelian Cayley graph spectrally, and it confirms that $O(\log |G|/\epsilon^2)$ generators suffice, with this bound being tight.

Spectral sparsification is a central tool in combinatorics and algorithmic graph theory, providing ways to compress graphs or algebraic objects while preserving spectral or cut properties. Classical sparsification techniques, such as Benczur-Karger cut sparsification and spectral sparsification via effective resistances or leverage scores, yield subgraphs that approximate the Laplacian of the original. However, these techniques typically lose the algebraic structure of Cayley graphs when applied, motivating the search for sparsifiers that remain Cayley graphs for the same group.

Abelian Cayley sparsification has applications in graph algorithms, coding theory (via the connection to $\mathbb{F}_2$-linear codes), and constraint satisfaction problems. Previous works have established upper bounds for the generator set size, but with suboptimal polylogarithmic losses, or only for specific group families.

Main Results and Technical Overview

The principal theorem asserts that for any finite abelian group $G$ and any Cayley graph $Cay(G, S, w)$, there exists a spectral sparsifier $Cay(G, S', w')$ with $|S'| = O(\log |G|/\epsilon^2)$ such that the Laplacian $L'$ of the sparsifier satisfies $(1-\epsilon)L \preceq L' \preceq (1+\epsilon)L$ for all $\epsilon \in (0,1)$. This is the tightest possible bound: the dependence on $\log |G|$ cannot be improved.

The result matches and strengthens prior bounds. For $\mathbb{F}_2$0, it yields $\mathbb{F}_2$1 generator sets, removing polylogarithmic losses found in previous works ([KhannaPS24]). Additionally, it generalizes Alon-Roichman's result for the complete Cayley graph with $\mathbb{F}_2$2 generators to arbitrary symmetric sets $\mathbb{F}_2$3.

The construction is randomized with runtime $\mathbb{F}_2$4, exploiting the symmetry of abelian groups and character theory. The sparsification does not depend on the weights and holds for weighted Cayley graphs.

Proof Structure and Novel Techniques

A key technical innovation is reducing Cayley sparsification to a volumetric analysis of a well-chosen convex body—the "sparsification polytope"—whose structure is governed by the symmetry of group characters. The proof builds on volume arguments from discrepancy theory ([ReisR26]), showing that as long as the generator set $\mathbb{F}_2$5 is large enough, it is possible to shrink its size by finding partial colorings that remove a constant fraction of generators while incurring only a small spectral error.

The process is iterative: at each step, a centrally symmetric convex body defined by spectral constraints ($\mathbb{F}_2$6) is analyzed, and a randomized algorithm finds a coloring that makes many coordinates $\mathbb{F}_2$7, effectively removing corresponding generators. The crux is an elementary symmetry argument, leveraging the character group structure of abelian groups to show that the convex body has sufficiently large volume, guaranteeing the existence of partial colorings. The repeated application of Minkowski-type theorems enables a reduction to an optimal generator set size.

The proof demonstrates that for $\mathbb{F}_2$8, the convex body $\mathbb{F}_2$9 contains many points with coordinates $G$0 and thus supports the iterative sparsification procedure. The volumetric lower bound is tight, and a contradiction argument shows that the dependence on $G$1 cannot be improved for abelian groups ([BasuKLM26]).

Implications and Applications

The sharp sparsification bounds have several immediate and broad implications:

• Coding Theory: For $G$2-linear codes, the result implies $G$3-sized code sparsifiers, a notable improvement over [KhannaPS24]. This demonstrates an equivalence between code sparsification and Cayley sparsification for $G$4.
• Ramanujan Bound Generalization: The theorem generalizes classical expander constructions such as Alon-Roichman to arbitrary symmetric sets and eigenvalue spectra, impacting pseudorandomness, complexity, and expander graph theory.
• Algorithmic Applications: Spectral sparsifiers are crucial for efficient Laplacian system solvers, streaming algorithms, sketching, and cut-preserving reductions ([SpielmanT11], [AhnGM12b]). Maintaining the algebraic structure allows for efficient computations and preserves properties required in coding and Markov chain analysis.
• Connections to Complexity and Derandomization: By preserving group structure, these sparsifiers can be applied to issues in derandomization, pseudorandom generators, and property testing in group-centric CSPs ([JalanM21]).

Open Problems and Future Directions

The construction's runtime is $G$5, which is infeasible for large groups like $G$6 for large $G$7. Deterministic polynomial-time algorithms achieving the optimal generator size remain open. The tradeoff between explicit constructions, derandomization, and optimality is of central interest, linking to $G$8-biased sets and expanders.

Further, the extension of these tight bounds to non-abelian groups, hypergraphs, or more general algebraic structures is an active area, as are applications to quantum Hamiltonian complexity—where sparsification impacts simulation and quantum coding—as well as to Boolean function analysis on high-dimensional expanders.

Conclusion

This work precisely resolves the abelian Cayley sparsification problem, showing that $G$9 generators suffice for spectral sparsification, and that this bound is optimal, both for weighted and unweighted graphs. The approach, grounded in volumetric convex geometry and character symmetry, enables powerful spectral reduction while preserving algebraic structure. The results advance understanding in graph algorithms, coding theory, and algebraic sparsification, and their implications extend to algorithmic efficiency, complexity theory, and pseudorandom constructions. Resolving algorithmic optimality and extensions beyond abelian groups represents a promising direction for future research.