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# Popular Matchings

Abraham, David J and Irving, Robert W and Kavitha, Telikepalli and Mehlhorn, Kurt (2007) Popular Matchings. In: SIAM Journal on Computing, 37 (4). pp. 1030-1045.

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Official URL: http://dl.acm.org/citation.cfm?id=1070491

## 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 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 of the preference lists. For the general case in which preference lists may contain ties, we give an $O(\sqrt{n}m)$ time algorithm.

Item Type: Journal Article Copyright of this article belongs to Society for Industrial and Applied Mathematics. matchings;bipartite graphs;one-sided preference lists. Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation) 03 Jul 2008 22 Feb 2012 10:08 http://eprints.iisc.ernet.in/id/eprint/14509

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