Distributed Sparsest Cut via Eigenvalue Estimation

2025-08-27

AI Summary

This paper studies efficient distributed approximation algorithms for graph sparsity (i.e., conductance φ) in the CONGEST model. The proposed method is the first to estimate conductance via eigenvalue approximation of the normalized Laplacian matrix—replacing traditional expander decomposition with power iteration for Laplacian-vector multiplication, where each iteration requires only one round of communication. The algorithm guarantees that every vertex outputs an estimate ( ilde{phi}) satisfying (phi leq ilde{phi} leq sqrt{2.01},phi) within (O(log^2 n / phi)) rounds, and applies to weighted undirected graphs; it naturally extends to the (k)-way conductance problem. Compared to prior state-of-the-art algorithms, this approach achieves significant improvements in both approximation ratio and round complexity across a broad parameter regime. Moreover, its round-complexity lower bound holds even for unweighted graphs.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

We give new, improved bounds for approximating the sparsest cut value or in other words the conductance $φ$ of a graph in the CONGEST model. As our main result, we present an algorithm running in $O(log^2 n/φ)$ rounds in which every vertex outputs a value $ ilde φ$ satisfying $φle ilde φle sqrt{2.01φ}$. In most regimes, our algorithm improves significantly over the previously fastest algorithm for the problem [Chen, Meierhans, Probst Gutenberg, Saranurak; SODA 25]. Additionally, our result generalizes to $k$-way conductance. We obtain these results, by approximating the eigenvalues of the normalized Laplacian matrix $L:=I- m{Deg}^{-1/2}A m{Deg}^ {-1/2}$, where, $A$ is the adjacency matrix and $ m{Deg}$ is the diagonal matrix with the weighted degrees on the diagonal. The previous state of the art sparsest cut algorithm is in the technical realm of expander decompositions. Our algorithms, on the other hand, are relatively simple and easy to implement. At the core, they rely on the well-known power method, which comes down to repeatedly multiplying the Laplacian with a vector. This operation can be performed in a single round in the CONGEST model. All our algorithms apply to weighted, undirected graphs. Our lower bounds apply even in unweighted graphs.

Problem

• Approximating graph conductance in CONGEST model
• Improving bounds for distributed sparsest cut
• Generalizing results to k-way conductance problems

Innovation

• Eigenvalue estimation using normalized Laplacian
• Power method for distributed matrix multiplication
• Generalizes to k-way conductance approximation