Unpopularity Factor in the Marriage and Roommates Problems

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 with opposite genders can be matched. For a pair of matchings X and Y , let φ(X, Y ) denote the number of people who prefer the person they get matched by X to the 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 the 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.

Popular matchings with weighted voters

“…For a loop e = {v, v}, they set vote M (e) = −w(v) if v is matched in M and vote M (e) = 0 otherwise. Theorem 4 (Ruangwises and Itoh [34]). M is popular if and only if vote M ( M ′ ) ≤ 0 for every matching M ′ in G, where G is the input graph G augmented with a loop {v, v} for every vertex v ∈ V , as defined in Section 2.2.2.…”

“…Ruangwises and Itoh [34] extended Theorem 2 to instances with weighted voters. They redefined the vote of…”

“…Itoh and Watanabe [22] studied the existence probability of a popular matching in random house allocation instances with vertex weights. Ruangwises and Itoh [34] designed an algorithm to compute the approximability measure called unpopularity factor for a given matching. As a byproduct, they also developed a polynomial algorithm to verify the popularity of a given matching, even in non-bipartite instances.…”

“…Whenever we will use vote M u (v) in the rest of the paper, this will refer to this vote function (and not the special unweighted case introduced in Section 2.2.2). For an edge e = {u, v}, Ruangwises and Itoh [34] defined vote M (e) := vote M u (v) + vote M v (u). For a loop e = {v, v}, they set vote M (e) = −w(v) if v is matched in M and vote M (e) = 0 otherwise.…”