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# Bounded Unpopularity Matchings

Huang, Chien-Chung and Kavitha, Telikepalli and Michail, Dimitrios and Nasre, Meghana (2011) Bounded Unpopularity Matchings. In: Algorithmica, 61 (3). pp. 738-757.

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## Abstract

We investigate the following problem: given a set of jobs and a set of people with preferences over the jobs, what is the optimal way of matching people to jobs? Here we consider the notion of popularity. A matching M is popular if there is no matching M' such that more people prefer M' to M than the other way around. Determining whether a given instance admits a popular matching and, if so, finding one, was studied by Abraham et al. (SIAM J. Comput. 37(4):1030-1045, 2007). If there is no popular matching, a reasonable substitute is a matching whose unpopularity is bounded. We consider two measures of unpopularity-unpopularity factor denoted by u(M) and unpopularity margin denoted by g(M). McCutchen recently showed that computing a matching M with the minimum value of u(M) or g(M) is NP-hard, and that if G does not admit a popular matching, then we have u(M) >= 2 for all matchings M in G. Here we show that a matching M that achieves u(M) = 2 can be computed in O(m root n) time (where m is the number of edges in G and n is the number of nodes) provided a certain graph H admits a matching that matches all people. We also describe a sequence of graphs: H = H(2), H(3), ... , H(k) such that if H(k) admits a matching that matches all people, then we can compute in O(km root n) time a matching M such that u(M) <= k - 1 and g(M) <= n(1 - 2/k). Simulation results suggest that our algorithm finds a matching with low unpopularity in random instances.

Item Type: Journal Article Copyright of this article belongs to Springer. Matching with preferences;Popularity;Approximation algorithms Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation) 16 Sep 2011 06:58 16 Sep 2011 06:59 http://eprints.iisc.ernet.in/id/eprint/40394

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