Unpopularity Factor in the Marriage and Roommates Problems - Ruangwises, Suthee; Itoh, Toshiya (Theory of computing systems, 2021)
Given a set A of n people, with each person having a preference list that ranks a subset of A as his/her acceptable partners in order of preference, we consider the Roommates Problem ( rp ) and the Marriage Problem ( mp ) of matching people with their partners. In rp there is no further restriction, while in mp only people of opposite genders can be acceptable partners. For a pair of matchings X and Y , let ϕ ( X , Y ) denote the number of people who prefer a person they get matched by X to a person they get matched by Y , and define an unpopularity factor u ( M ) of a matching M to be the maximum ratio ϕ ( M ′ , M ) / ϕ ( M , M ′ ) among all other possible matchings M ′ . In this paper, we develop an algorithm to compute the unpopularity factor of a given matching in O ( m n log 2 n ) time for rp and in O ( m 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 a weighted setting where people are given different voting weights and show that our algorithm can be slightly modified to support that setting with the same running time.