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 $$\phiG(S) := \frac{w(E(S,\bar{S}))}{\min \set{w(S), w(\bar{S})}}, $$ 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 $\set{P1, \ldots, Pk}$ of the vertex set so as to minimize $$ \phik(\set{P1, \ldots, Pk}) := \maxi \phiG(Pi). $$ 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 $O{\varepsilon}(\sqrt{\log n \log k})$ times $OPT$. We also study balanced versions of this problem.