Lecture Notes on the ARV Algorithm for Sparsest Cut

One of the landmarks in approximation algorithms is the O(√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/√log n) to each other if measured in terms of |· |22. 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.