Min-Cost Popular Matchings.

We consider the problem of matching a set of applicants to a set of posts , where each applicant has a preference list , ranking a nonempty subset of posts in order of preference, possibly involving ties. We say that a matching $M$ is popular if there is no matching $M'$ such that the number of applicants preferring $M'$ to $M$ exceeds the number of applicants preferring $M$ to $M'$. In this paper, we give the first polynomial-time algorithms to determine if an instance admits a popular matching and to find a largest such matching, if one exists. For the special case in which every preference list is strictly ordered (i.e., contains no ties), we give an $O(n + m)$ time algorithm, where $n$ is the total number of applicants and posts and $m$ is the total length of all of the preference lists. For the general case in which preference lists may contain ties, we give an $O(\sqrt{n}m)$ time algorithm.