The Popular Matching Problem in a (3, 3)-Hypergraph with One-sided Preferences.

In this work we investigate the problem of popular matching in a 3-uniform 3-partite hypergraph, where the first set contains a set of agents and the second and the third set contain different class of items respectively. Agents have a preference list for hyperedges, whereas the members of the other sets do not have preferences. Each agent votes in favor of a preferred matching. A matching M is called popular if there does not exist a matching M′ such that the number of agents that prefer M′ is more than the number of agents that prefer M. This paper provides a characterization of popular matching in a 3-uniform 3-partite hypergraph and shows that the problem of deciding whether a given 3-uniform 3-partite hypergraph has a popular matching is NP-hard. We also prove the NP-hardness of popular matching in a k-uniform k-partite hypergraph (k > 3). Assuming that a given 3-uniform 3-partite hypergraph admits at least one popular matching, we compare the cardinality of the maximum size popular matching (denoted as M ^∗ ) with a maximum matching (denoted as M max ) in the 3-uniform 3-partite hypergraph and show that $\left| {{M^{\ast}}} \right| \geq \frac{2}{3}\left| {{M_{\max }}} \right|$.