  Cuts in graphs are a fundamental object of study, and play a central role in
the study of graph algorithms. The problem of sparsifying a graph while
approximately preserving its cut structure has been extensively studied and has
many applications. In a seminal work, Bencz\'ur and Karger (1996) showed that
given any $n$-vertex undirected weighted graph $G$ and a parameter $\varepsilon
\in (0,1)$, there is a near-linear time algorithm that outputs a weighted
subgraph $G'$ of $G$ of size $\tilde{O}(n/\varepsilon^2)$ such that the weight
of every cut in $G$ is preserved to within a $(1 \pm \varepsilon)$-factor in
$G'$. The graph $G'$ is referred to as a {\em $(1 \pm \varepsilon)$-approximate
cut sparsifier} of $G$.
  A natural question is if such cut-preserving sparsifiers also exist for
hypergraphs. Kogan and Krauthgamer (2015) initiated a study of this question
and showed that given any weighted hypergraph $H$ where the cardinality of each
hyperedge is bounded by $r$, there is a polynomial-time algorithm to find a $(1
\pm \varepsilon)$-approximate cut sparsifier of $H$ of size
$\tilde{O}(\frac{nr}{\varepsilon^2})$. Since $r$ can be as large as $n$, in
general, this gives a hypergraph cut sparsifier of size
$\tilde{O}(n^2/\varepsilon^2)$, which is a factor $n$ larger than the
Bencz\'ur-Karger bound for graphs. It has been an open question whether or not
Bencz\'ur-Karger bound is achievable on hypergraphs. In this work, we resolve
this question in the affirmative by giving a new polynomial-time algorithm for
creating hypergraph sparsifiers of size $\tilde{O}(n/\varepsilon^2)$.
