  The expansion of a hypergraph, a natural extension of the notion of expansion
in graphs, is defined as the minimum over all cuts in the hypergraph of the
ratio of the number of the hyperedges cut to the size of the smaller side of
the cut. We study the Hypergraph Small Set Expansion problem, which, for a
parameter $\delta \in (0,1/2]$, asks to compute the cut having the least
expansion while having at most $\delta$ fraction of the vertices on the smaller
side of the cut. We present two algorithms. Our first algorithm gives an
$\tilde O(\delta^{-1} \sqrt{\log n})$ approximation. The second algorithm finds
a set with expansion $\tilde O(\delta^{-1}(\sqrt{d_{\text{max}}r^{-1}\log r\,
\phi^*} + \phi^*))$ in a $r$--uniform hypergraph with maximum degree
$d_{\text{max}}$ (where $\phi^*$ is the expansion of the optimal solution).
Using these results, we also obtain algorithms for the Small Set Vertex
Expansion problem: we get an $\tilde O(\delta^{-1} \sqrt{\log n})$
approximation algorithm and an algorithm that finds a set with vertex expansion
$O\left(\delta^{-1}\sqrt{\phi^V \log d_{\text{max}} } + \delta^{-1}
\phi^V\right)$ (where $\phi^V$ is the vertex expansion of the optimal
solution).
  For $\delta=1/2$, Hypergraph Small Set Expansion is equivalent to the
hypergraph expansion problem. In this case, our approximation factor of
$O(\sqrt{\log n})$ for expansion in hypergraphs matches the corresponding
approximation factor for expansion in graphs due to ARV.
