An $O\left(\sqrt{n}\right)$-Approximation Algorithm for Directed Sparsest Cut

We give an $O\left(\sqrt{n}\right)$-approximation algorithm for the Sparsest Cut Problem on directed graphs. A naïve reduction from Sparsest Cut to Minimum Multicut would only give an approximation ratio of $O(\sqrt{n}logD)$, where $D$ is the sum of the demands. We obtain the improvement using a novel LP-rounding method for fractional Sparsest Cut, the dual of Maximum Concurrent Flow.

Keywords: approximation algorithms, directed graphs, sparsest cut, multicommodity flow.