  Given a graph $G$, the sparsest-cut problem asks to find the set of vertices
$S$ which has the least expansion defined as $$\phi_G(S) :=
\frac{w(E(S,\bar{S}))}{\min \set{w(S), w(\bar{S})}}, $$ 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 $\set{P_1, \ldots, P_k}$
of the vertex set so as to minimize $$ \phi_k(\set{P_1, \ldots, P_k}) := \max_i
\phi_G(P_i). $$ 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.
