Popular matchings: structure and algorithms

Eric McDermid () and Robert W. Irving ()
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Eric McDermid: University of Glasgow
Robert W. Irving: University of Glasgow

Journal of Combinatorial Optimization, 2011, vol. 22, issue 3, No 4, 339-358

Abstract: Abstract An instance of the popular matching problem (POP-M) consists of a set of applicants and a set of posts. Each applicant has a preference list that strictly ranks a subset of the posts. A matching M of applicants to posts is popular if there is no other matching M′ such that more applicants prefer M′ to M than prefer M to M′. Abraham et al. (SIAM J. Comput. 37:1030–1045, 2007) described a linear time algorithm to determine whether a popular matching exists for a given instance of POP-M, and if so to find a largest such matching. A number of variants and extensions of POP-M have recently been studied. This paper provides a characterization of the set of popular matchings for an arbitrary POP-M instance in terms of a structure called the switching graph, a directed graph computable in linear time from the preference lists. We show that the switching graph can be exploited to yield efficient algorithms for a range of associated problems, including the counting and enumeration of the set of popular matchings, generation of a popular matching uniformly at random, finding all applicant-post pairs that can occur in a popular matching, and computing popular matchings that satisfy various additional optimality criteria. Our algorithms for computing such optimal popular matchings improve those described in a recent paper by Kavitha and Nasre (Proceedings of MATCH-UP: Matching Under Preferences—Algorithms and Complexity, 2008).

Keywords: Matchings; Bipartite graphs; One-sided preference lists (search for similar items in EconPapers)
Date: 2011
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Citations: View citations in EconPapers (8)

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DOI: 10.1007/s10878-009-9287-9

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