Unpopularity Factor in the Marriage and Roommates Problems

TL;DR

This paper introduces an efficient algorithm to compute the unpopularity factor of matchings in the Roommates and Marriage Problems, extending to weighted preferences, which measures how unpopular a matching is among participants.

Contribution

The paper develops a novel algorithm for calculating the unpopularity factor in RP and MP, with improved time complexity, and generalizes it to weighted preferences.

Findings

Algorithm runs in O(m√n log^2 n) for RP and O(m√n log n) for MP.

Supports weighted preferences with similar efficiency.

Provides a new metric for evaluating matchings' popularity.

Abstract

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 $\phi(X,Y)$ 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 $\phi(M',M) / \phi(M,M')$ among all other possible matchings $M'$. In this paper, we develop an algorithm to compute the unpopularity factor of a given matching in $O(m\sqrt{n}\log^2 n)$ 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.