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 factoru(M) of a matching M to be the maximum ratio \(\phi (M^{\prime },M) / \phi (M,M^{\prime })\) among all other possible matchings \(M^{\prime }\). In this paper, we develop an algorithm to compute the unpopularity factor of a given matching 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 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.

给定一组一的ñ人，有每个人的喜好列表是排名的一个子集，一个为他/她可以接受的合作伙伴按优先顺序排列，我们认为室友问题（RP）和婚姻问题（MP匹配的人）和他们的伙伴。在rp中没有进一步的限制，而在mp中只有异性可以成为可接受的伴侣。对于X和Y的一对匹配，令ϕ（X，Y）表示更喜欢X匹配的人比Y匹配的人的人数，并将匹配M的不受欢迎度u（M）定义为最大比率\（\ phi（M ^ { \ prime}，M）/ \ phi（M，M ^ {\ prime}）\）等所有可能的匹配项\（M ^ {\ prime} \）。在本文中，我们开发的算法来计算在给定的匹配的不受欢迎因子\（O（M \ SQRT {N} \日志^ {2} N）\）时间RP和\（O（M \ SQRT {n} \ log n）\）mp的时间，其中m是人们的偏好列表的总长度。我们还将不受欢迎因素的概念推广到加权设置中，在加权设置中人们被赋予不同的投票权重，并表明我们的算法可以稍加修改以在相同的运行时间下支持该设置。