Two-sided profile-based optimality in the stable marriage problem
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We study the problem of finding "fair" stable matchings in the Stable Marriage problem with Incomplete lists (SMI). In particular, we seek stable matchings that are optimal with respect to profile, which is a vector that indicates the number of agents who have their first-, second-, third-choice partner, etc. In a rank maximal stable matching, the maximum number of agents have their first-choice partner, and subject to this, the maximum number of agents have their second-choice partner, etc., whilst in a generous stable matching M, the minimum number of agents have their dth-choice partner, and subject to this, the minimum number of agents have their (d-1)th-choice partner, etc., where d is the maximum rank of an agent's partner in M. Irving et al. presented an O(n^5 n) algorithm for finding a rank-maximal stable matching, which can be adapted easily to the generous stable matching case, where n is the number of men / women. An O(n^4.5) algorithm for the rank-maximal stable problem was later given by Feder. However these approaches involve the use of weights that are in general exponential in n, potentially leading to inaccuracies or memory issues upon implementation. In this paper we present an O(n^5 n) algorithm for finding a rank-maximal stable matching using an approach that involves weights that are polynomially-bounded in n. We show how to adapt our algorithm for the generous case to run in O(n^2d^3 n) time. Additionally we conduct an empirical evaluation to compare various measures over many different types of "fair" stable matchings, including rank-maximal, generous, egalitarian, sex-equal and median stable matchings. In particular, we observe that a generous stable matching is typically considerably closer than a rank-maximal stable matching in terms of the egalitarian and sex-equality optimality criteria.
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