Lecture Notes on the ARV Algorithm for Sparsest Cut

One of the landmarks in approximation algorithms is theO( √ logn)-approximation algorithm for theUniform 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 randomhyperplane approachwill find two large sets ofΘ(n)many nodes each that have a distance ofΘ(1/ √ logn) to each other if measured in terms of ‖ ·‖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.