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 [EQUATION] 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 {P1, ..., Pk} of the vertex set so as to minimize [EQUATION] Our main result is a polynomial time bi-criteria approximation algorithm which outputs a (1 -- ε)k-partition of the vertex set such that each piece has expansion at most [EQUATION] times OPT. We also study balanced versions of this problem.