In the "expansion profile" problem and the "small-set expander" problem we are interested in the following question: given a graph and a parameter k, find the subset of k or fewer vertices with the smallest expansion.

Using the "evolving sets" algorithm of Anderson and Peres, we provide the following approximation guarantee: if there is a set of size at most k and expansion epsilon, we can find a set of size at most k*n^(1/c) and expansion at most O( (c * epsilon)^(1/2) ). Furthermore, the running time of the algorithm is nearly-linear in the size of the output set. This is the first algorithm for the expansion profile problem that does not lose factors of (log n)^(Omega(1)) in the expansion.

This lecture describes joint work with Shayan Oveis Gharan.