Lecture Notes on the ARV Algorithm for Sparsest Cut

TL;DR

This paper provides detailed lecture notes on the ARV algorithm for the Uniform Sparsest Cut problem, including a simplified proof of the Structure Theorem using expected maxima over neighborhoods.

Contribution

It offers a clearer, more accessible explanation of the ARV algorithm and a simplified proof of key theoretical results in approximation algorithms.

Findings

Simplified proof of the Structure Theorem using expected maxima.

Detailed explanation of the ARV algorithm for Sparsest Cut.

Clarification of the hyperplane rounding technique.

Abstract

One of the landmarks in approximation algorithms is the $O(\sqrt{\log n})$-approximation algorithm for the Uniform Sparsest Cut problem by Arora, Rao and Vazirani from 2004. The algorithm is based on a semidefinite program that finds an embedding of the nodes respecting the triangle inequality. Their core argument shows that a random hyperplane approach will find two large sets of $\Theta(n)$ many nodes each that have a distance of $\Theta(1/\sqrt{\log n})$ to each other if measured in terms of $\|\cdot \|_2^2$. Here we give a detailed set of lecture notes describing the algorithm. For the proof of the Structure Theorem we use a cleaner argument based on expected maxima over $k$-neighborhoods that significantly simplifies the analysis.