Computing at Glasgow University
Paper ID: 8590

Popular Matchings
Abraham,D.J. Irving,R.W. Kavitha,T. Mehlhorn,K.

Publication Type: Journal
Appeared in: SIAM Journal on Computing, Volume 37
Page Numbers : 1030 - 1045
Publisher: Society for Industrial and Applied Mathematics
Year: 2007

We consider the problem of matching a set of applicants to a set of posts, where each applicant has a preference list, ranking a non-empty 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 the preference lists. For the general case in which preference lists may contain ties, we give an O(\sqrt{n}m) time algorithm, and show that the problem has equivalent time complexity to the maximum-cardinality bipartite matching problem.

Keywords: Matching, polynomial-time algorithm

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