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
ISBN/ISSN:
Abstract:
We consider the problem of matching a set of applicants to a set of posts, where each applicant has a preference list, ranking a nonempty 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 polynomialtime 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 maximumcardinality bipartite matching problem.
Keywords: Matching, polynomialtime algorithm
