Weighted popular matchings

Abstract

We study the problem of assigning applicants to jobs. Each applicant has a weight and provides a preference list, which may contain tics, ranking a subset of the jobs. An applicant x may prefer one matching over the other (or be indifferent between them, in case of a tie) based on the jobs x gets in the two matchings and x's personal preference. A matching M is popular if there is no other matching M′ such that the weight of the applicants who prefer M′ over M exceeds the weight of those who prefer M over M′. We present two algorithms to find a popular matching, or in case none exists, to establish so, For the case of strict preferences we develop an O(n + m) time algorithm. When ties are allowed a more involved algorithm solves the problem in O(min(k√n, n)m) time, where k is the number of distinct weights the applicants are given. © Springer-Verlag Berlin Heidelberg 2006.