Popular Matchings with Two-Sided Preferences and One-Sided Ties

We are given a bipartite graph G = (A ∪ B,E) where each vertex has a preference list ranking its neighbors: in particular, every a ∈ A ranks its neighbors in a strict order of preference, whereas the preference lists of b ∈ 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 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 ∈ 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 ∈ B puts all its neighbors into a single tie. That is, all neighbors of b are tied in b’s list and and b desires to be matched to any of them. Our main result is an O(n) algorithm (where n = |A ∪ B|) 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.