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Given a set $A$ of $n$ people, with each person having a preference list thatranks a subset of $A$ as his/her acceptable partners in order of preference, weconsider the Roommates Problem (RP) and the Marriage Problem (MP) of matchingpeople with their partners. In RP there is no further restriction, while in MPonly people of opposite genders can be acceptable partners. For a pair ofmatchings $X$ and $Y$, let $\phi(X,Y)$ denote the number of people who prefer aperson they get matched by $X$ to a person they get matched by $Y$, and definean unpopularity factor $u(M)$ of a matching $M$ to be the maximum ratio$\phi(M',M) / \phi(M,M')$ among all other possible matchings $M'$. In thispaper, we develop an algorithm to compute the unpopularity factor of a givenmatching in $O(m\sqrt{n}\log^2 n)$ time for RP and in $O(m\sqrt{n}\log n)$ timefor MP, where $m$ is the total length of people's preference lists. We alsogeneralize the notion of unpopularity factor to a weighted setting where peopleare given different voting weights and show that our algorithm can be slightlymodified to support that setting with the same running time.

Unpopularity Factor in the Marriage and Roommates Problems