Sparsest cut and eigenvalue multiplicities on low degree Abelian Cayley graphs
Tommaso d'Orsi, Chris Jones, Jake Ruotolo, Salil Vadhan, Jiyu Zhang
TL;DR
This work develops a higher-order spectral framework for the Sparsest Cut problem, introducing the solution-dimension parameter $SD_{\\varepsilon,c}(G)$ to quantify when eigenspace enumeration plus cut-improvement yields near-optimal cuts. It proves a simple Cut Improvement algorithm and ties runtime to $SD_{\\varepsilon,c}(G)$, showing that for low-degree Abelian Cayley graphs the sparsest cut can be approximated arbitrarily well in time $n^{O(1)} \cdot \exp((d/\\varepsilon)^{O(d)})$, with PTAS for very small degrees. A key technical contribution is a tight bound on eigenvalue multiplicity $mul_{\\lambda_2}(G) \\le 2^{O(d)}$ for Abelian Cayley graphs, together with a robust link between sparse cuts and the low-eigenvalue subspace that yields a small-dimensional embedding for near-optimal cuts. The paper further establishes that the sparse-cut structure can be captured in the first $d^{O(d)}$ eigenvectors, and it provides constructive results for Cayley graphs over $\\mathbb{Z}_p^k$, including a polynomial-time $O(p)$-approximation for sparsest cut. Overall, these results illuminate the spectral–combinatorial landscape of sparsest cut on structured graphs and suggest efficient algorithms in regimes where $d$ is small relative to $n$.
Abstract
Whether or not the Sparsest Cut problem admits an efficient $O(1)$-approximation algorithm is a fundamental algorithmic question with connections to geometry and the Unique Games Conjecture. Revisiting spectral algorithms for Sparsest Cut, we present a novel, simple algorithm that combines eigenspace enumeration with a new algorithm for the Cut Improvement problem. The runtime of our algorithm is parametrized by a quantity that we call the solution dimension $\text{SD}_\varepsilon(G)$: the smallest $k$ such that the subspace spanned by the first $k$ Laplacian eigenvectors contains all but $\varepsilon$ fraction of a sparsest cut. Our algorithm matches the guarantees of prior methods based on the threshold-rank paradigm, while also extending beyond them. To illustrate this, we study its performance on low degree Cayley graphs over Abelian groups -- canonical examples of graphs with poor expansion properties. We prove that low degree Abelian Cayley graphs have small solution dimension, yielding an algorithm that computes a $(1+\varepsilon)$-approximation to the uniform Sparsest Cut of a degree-$d$ Cayley graph over an Abelian group of size $n$ in time $n^{O(1)}\cdot\exp(d/\varepsilon)^{O(d)}$. Along the way to bounding the solution dimension of Abelian Cayley graphs, we analyze their sparse cuts and spectra, proving that the collection of $O(1)$-approximate sparsest cuts has an $\varepsilon$-net of size $\exp(d/\varepsilon)^{O(d)}$ and that the multiplicity of $λ_2$ is bounded by $2^{O(d)}$. The latter bound is tight and improves on a previous bound of $2^{O(d^2)}$ by Lee and Makarychev.