  Haxell's condition is a natural hypergraph analog of Hall's condition, which
is a well-known necessary and sufficient condition for a bipartite graph to
admit a perfect matching. That is, when Haxell's condition holds it forces the
existence of a perfect matching in the bipartite hypergraph. Unlike in graphs,
however, there is no known polynomial time algorithm to find the hypergraph
perfect matching that is guaranteed to exist when Haxell's condition is
satisfied.
  We prove the existence of an efficient algorithm to find perfect matchings in
bipartite hypergraphs whenever a stronger version of Haxell's condition holds.
Our algorithm can be seen as a generalization of the classical Hungarian
algorithm for finding perfect matchings in bipartite graphs. The techniques we
use to achieve this result could be of use more generally in other
combinatorial problems on hypergraphs where disjointness structure is crucial,
e.g. Set Packing.
