Sparsest cut and eigenvalue multiplicities on low degree Abelian Cayley graphs

2024-12-22

arXiv.org

AI Summary

The sparsest cut problem’s O(1)-approximation remains a foundational challenge bridging geometry, spectral graph theory, and the Unique Games Conjecture. This work focuses on low-degree Abelian Cayley graphs and introduces the novel notion of *cut dimension* (CDₐ), proving it is bounded by 2ᴼ⁽ᵈ⁾ for d-regular graphs. Leveraging this, we design the first polynomial-time (1+ε)-approximate spectral algorithm: using Fourier analysis to characterize relevant eigenspaces, enumerating low-dimensional cut structures, and applying Cut Improvement, we construct an ε-net for sparse cuts of size exp(d/ε)ᴼ⁽ᵈ⁾. The algorithm runs in nᴼ⁽¹⁾·exp(d/ε)ᴼ⁽ᵈ⁾ time. Crucially, our analysis tightens the upper bound on the multiplicity of the second smallest Laplacian eigenvalue λ₂ from 2ᴼ⁽ᵈ²⁾ to 2ᴼ⁽ᵈ⁾. This significantly extends the applicability of spectral methods to graphs with poor expansion properties.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 cut dimension $ ext{CD}_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 cut 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 $n^{O(1)}cdotexp(d/varepsilon)^{O(d)}$. Along the way to bounding the cut 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 $lambda_2$ 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.

Problem

• Efficient approximation algorithm for Sparsest Cut problem
• Analyzing cut dimension on low-degree Abelian Cayley graphs
• Improving eigenvalue multiplicity bounds for graph spectra

Innovation

• Eigenspace enumeration with Cut Improvement algorithm
• Runtime parameterized by cut dimension CD_ε(G)
• Analyzes sparse cuts and spectra of Abelian Cayley graphs