Popular matchings with two-sided preferences and one-sided ties

We are given a bipartite graph G - (A boolean OR B, E) where each vertex has a preference list ranking its neighbors: in particular, every a is an element of A ranks its neighbors in a strict order of preference, whereas the preference lists of b is an element of B may contain ties. A matching M is popular if there is no matching M ' such that the number of vertices that prefer M ' to M exceeds the number of vertices that prefer M to M '. We show that the problem of deciding whether G admits a popular matching or not is NP-hard. This is the case even when every b is an element of B either has a strict preference list or puts all its neighbors into a single tie. In contrast, we show that the problem becomes polynomially solvable in the case when each b is an element of B puts all its neighbors into a single tie. That is, all neighbors of b are tied in b 's list and b desires to be matched to any of them. Our main result is an O(n(2)) algorithm (where n=vertical bar A boolean OR B vertical bar) for the popular matching problem in this model. Note that this model is quite different from the model where vertices in B have no preferences and do not care whether they are matched or not