Popular matchings in the marriage and roommates problems

Popular matchings have recently been a subject of study in the context of the so-called House Allocation Problem, where the objective is to match applicants to houses over which the applicants have preferences. A matching M is called popular if there is no other matching M′ with the property that more applicants prefer their allocation in M′ to their allocation in M. In this paper we study popular matchings in the context of the Roommates Problem, including its special (bipartite) case, the Marriage Problem. We investigate the relationship between popularity and stability, and describe efficient algorithms to test a matching for popularity in these settings. We also show that, when ties are permitted in the preferences, it is NP-hard to determine whether a popular matching exists in both the Roommates and Marriage cases

Popular Matchings in Complete Graphs

Our input is a complete graph $G = (V,E)$ on $n$ vertices where each vertex has a strict ranking of all other vertices in $G$. Our goal is to construct a matching in $G$ that is popular. A matching $M$ is popular if $M$ does not lose a head-to-head election against any matching $M'$, where each vertex casts a vote for the matching in $\{M,M'\}$ where it gets assigned a better partner. The popular matching problem is to decide whether a popular matching exists or not. The popular matching problem in $G$ is easy to solve for odd $n$. Surprisingly, the problem becomes NP-hard for even $n$, as we show here.Comment: Appeared at FSTTCS 201

Popular Roommates in Simply Exponential Time

We consider the popular matching problem in a graph G = (V,E) on n vertices with strict preferences. A matching M is popular if there is no matching N in G such that vertices that prefer N to M outnumber those that prefer M to N. It is known that it is NP-hard to decide if G has a popular matching or not. There is no faster algorithm known for this problem than the brute force algorithm that could take n! time. Here we show a simply exponential time algorithm for this problem, i.e., one that runs in O^*(k^n) time, where k is a constant. We use the recent breakthrough result on the maximum number of stable matchings possible in such instances to analyze our algorithm for the popular matching problem. We identify a natural (also, hard) subclass of popular matchings called truly popular matchings and show an O^*(2^n) time algorithm for the truly popular matching problem

Popular Matchings in Complete Graphs

Our input is a complete graph G = (V,E) on n vertices where each vertex has a strict ranking of all other vertices in G. The goal is to construct a matching in G that is "globally stable" or popular. A matching M is popular if M does not lose a head-to-head election against any matching M\u27: here each vertex casts a vote for the matching in {M,M\u27} where it gets a better assignment. Popular matchings need not exist in the given instance G and the popular matching problem is to decide whether one exists or not. The popular matching problem in G is easy to solve for odd n. Surprisingly, the problem becomes NP-hard for even n, as we show here

Computational complexity of kk-stable matchings

We study deviations by a group of agents in the three main types of matching markets: the house allocation, the marriage, and the roommates models. For a given instance, we call a matching $k$-stable if no other matching exists that is more beneficial to at least $k$ out of the $n$ agents. The concept generalizes the recently studied majority stability. We prove that whereas the verification of $k$-stability for a given matching is polynomial-time solvable in all three models, the complexity of deciding whether a $k$-stable matching exists depends on $\frac{k}{n}$ and is characteristic to each model.Comment: SAGT 202

Matchings and Copeland's Method

Given a graph $G = (V,E)$ where every vertex has a weak ranking over its neighbors, we consider the problem of computing an optimal matching as per agent preferences. The classical notion of optimality in this setting is stability. However stable matchings, and more generally, popular matchings need not exist when $G$ is non-bipartite. Unlike popular matchings, Copeland winners always exist in any voting instance -- we study the complexity of computing a matching that is a Copeland winner and show there is no polynomial-time algorithm for this problem unless $\mathsf{P} = \mathsf{NP}$. We introduce a relaxation of both popular matchings and Copeland winners called weak Copeland winners. These are matchings with Copeland score at least $\mu/2$, where $\mu$ is the total number of matchings in $G$; the maximum possible Copeland score is $(\mu-1/2)$. We show a fully polynomial-time randomized approximation scheme to compute a matching with Copeland score at least $\mu/2\cdot(1-\varepsilon)$ for any $\varepsilon > 0$