  $ $We study the $d$-Uniform Hypergraph Matching ($d$-UHM) problem: given an
$n$-vertex hypergraph $G$ where every hyperedge is of size $d$, find a maximum
cardinality set of disjoint hyperedges. For $d\geq3$, the problem of finding
the maximum matching is NP-complete, and was one of Karp's 21
$\mathcal{NP}$-complete problems. In this paper we are interested in the
problem of finding matchings in hypergraphs in the massively parallel
computation (MPC) model that is a common abstraction of MapReduce-style
computation. In this model, we present the first three parallel algorithms for
$d$-Uniform Hypergraph Matching, and we analyse them in terms of resources such
as memory usage, rounds of communication needed, and approximation ratio. The
highlights include:
  $\bullet$ A $O(\log n)$-round $d$-approximation algorithm that uses $O(nd)$
space per machine.
  $\bullet$ A $3$-round, $O(d^2)$-approximation algorithm that uses
$\tilde{O}(\sqrt{nm})$ space per machine.
  $\bullet$ A $3$-round algorithm that computes a subgraph containing a
$(d-1+\frac{1}{d})^2$-approximation, using $\tilde{O}(\sqrt{nm})$ space per
machine for linear hypergraphs, and $\tilde{O}(n\sqrt{nm})$ in general.
