Abstract
Given a set A of n people, we consider the Roommates Problem (rp) and Marriage Problem (mp) where each person has a list that ranks a subset of A as his/her acceptable partner in order of preference. Ties among two or more people are allowed in the lists. In rp there is no further restriction, while in mp only people with opposite genders can be matched. For a pair of matchings X and Y, we say a person prefers X to Y if he/she prefers the person matched by X to the person matched by Y, and let \(\phi (X,Y)\) denote the number of people who prefer X to Y. Define an unpopularity factor u(M) of a matching M to be the maximum ratio \(\phi (M',M) / \phi (M,M')\) among all possible other matchings \(M'\). In this paper, we develop an algorithm to efficiently compute the unpopularity factor of a given matching. The algorithm runs in \(O(m\sqrt{n}\log ^2 n)\) time for rp and in \(O(m\sqrt{n}\log n)\) time for mp, where m is the total length of people’s preference lists. We also generalize the notion of unpopularity factor to the weighted setting where people are given different voting weights, and show that our algorithm can be slightly modified to support that setting as well with the same runtime.
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Ruangwises, S., Itoh, T. (2019). Unpopularity Factor in the Marriage and Roommates Problems. In: van Bevern, R., Kucherov, G. (eds) Computer Science – Theory and Applications. CSR 2019. Lecture Notes in Computer Science(), vol 11532. Springer, Cham. https://doi.org/10.1007/978-3-030-19955-5_29
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