  We initiate the study of approximating the largest induced expander in a
given graph $G$. Given a $\Delta$-regular graph $G$ with $n$ vertices, the goal
is to find the set with the largest induced expansion of size at least $\delta
\cdot n$. We design a bi-criteria approximation algorithm for this problem; if
the optimum has induced spectral expansion $\lambda$ our algorithm returns a
$\frac{\lambda}{\log^2\delta \exp(\Delta/\lambda)}$-(spectral) expander of size
at least $\delta n$ (up to constants).
  Our proof introduces and employs a novel semidefinite programming relaxation
for the largest induced expander problem. We expect to see further applications
of our SDP relaxation in graph partitioning problems. In particular, because of
the close connection to the small set expansion problem, one may be able to
obtain new insights into the unique games problem.
