Weighted popular matchings

Abstract

We study the problem of assigning jobs to applicants. Each applicant has a weight and provides a preference list , which may contain ties, ranking a subset of the jobs. An applicant x may prefer one matching to another (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 ′ to M exceeds the weight of those who prefer M to M ′. We present algorithms to find a popular matching, or if none exists, to establish so. For instances with strict preference lists, we give an O ( n + m time algorithm. For preference lists with ties, we give a more involved algorithm that solves the problem in O (min ( k √ n ;, n ) m ) time, where k is the number of distinct weights the applicants are given.