The Generalized Popular Condensation Problem

Abstract

AbstractIn this paper, we are concerned with the generalized popular condensation problem, which is an extension of the popular matching problem. An instance of the popular matching problem consists of a set of applicants \(A\), a set of posts \(P\), and a set of preference lists. According to the preference lists, the goal is to match each applicant with at most one unique post so that the resulting matching is “popular.” A matching \(M\) mapping from \(A\) to \(P\) is popular if there is no other matching \(M'\) such that more applicants prefer \(M'\) to \(M\) than prefer \(M\) to \(M'\). However, such a matching does not always exist. To fulfill the popular matching requirements, a possible manipulation is to neglect some applicants. Given that each applicant is appended with a cost for neglecting, the generalized popular condensation problem asks for a subset of applicants, with minimum sum of costs, whose deletion results in an instance admitting a popular matching. We prove that this problem is NP-hard, and it is also NP-hard to approximate to within a certain factor, assuming ties are involved in the preference lists. For the special case where the costs of all applicants are equal, we show that the problem can be solved in \(O(\sqrt{n}m)\) time, where \(n\) is the number of applicants and posts, and \(m\) is the total length of the preference lists.KeywordsBipartite GraphLocal ApplicantMaximum MatchSatisfying AssignmentPreference ListThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.