# Distributed Sparsest Cut via Eigenvalue Estimation

## TL;DR

This paper introduces a simple, efficient distributed algorithm in the CONGEST model for approximating the sparsest cut and conductance of a graph by estimating eigenvalues of the normalized Laplacian, improving over previous methods.

## Contribution

It presents a new distributed eigenvalue approximation algorithm for sparsest cut with better bounds and simplicity compared to prior expander decomposition approaches.

## Key findings

- Runs in $O(rac{ ext{log}^2 n}{ ext{phi}})$ rounds
- Achieves $ ilde{ ext phi}$ with $ extphi extless= ilde{ ext phi} extless= extsqrt{2.01 extphi}$
- Applicable to weighted, undirected graphs with lower bounds in unweighted cases.

## Abstract

We give new, improved bounds for approximating the sparsest cut value or in other words the conductance $\phi$ of a graph in the CONGEST model. As our main result, we present an algorithm running in $O(\log^2 n/\phi)$ rounds in which every vertex outputs a value $\tilde \phi$ satisfying $\phi \le \tilde \phi \le \sqrt{2.01\phi}$. 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 $L:=I-\rm{Deg}^{-1/2}A\rm{Deg}^ {-1/2}$, where, $A$ is the adjacency matrix and $\rm{Deg}$ 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.