Approximation Algorithm for Sparsest k-Partitioning

Given a graph G, the sparsest-cut problem asks to find the set of vertices S which has the least expansion defined as φG(S) := fracw(E(S,barS))min setw(S), w(barS), where w is the total edge weight of a subset. Here we study the natural generalization of this problem: given an integer k, compute a k-partition setP1, ldots, Pk of the vertex set so as to minimize φk(setP1, ldots, Pk) := maxi φG(Pi). Our main result is a polynomial time bi-criteria approximation algorithm which outputs a (1 - e)k-partition of the vertex set such that each piece has expansion at most Ovarepsilon(√log n log k) times OPT. We also study balanced versions of this problem.