Popular matchings

We consider the problem of matching a set of <i>applicants</i> to a set of <i>posts</i>, where each applicant has a <i>preference list</i>, ranking a non-empty subset of posts in order of preference, possibly involving ties. We say that a matching <i>M</i> is <i>popular</i> if there is no matching <i>M'</i> such that the number of applicants preferring <i>M'</i> to <i>M</i> exceeds the number of applicants preferring <i>M</i> to <i>M'</i>. 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 <i>O</i>(<i>n</i>+<i>m</i>) time algorithm, where <i>n</i> is the total number of applicants and posts, and <i>m</i> is the total length of all the preference lists. For the general case in which preference lists may contain ties, we give an <i>O</i>(√<i>nm</i>) time algorithm, and show that the problem has equivalent time complexity to the maximum-cardinality bipartite matching problem.