17 research outputs found
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 MⲠwith the property that more applicants prefer their allocation in MⲠto 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 on vertices where each vertex
has a strict ranking of all other vertices in . Our goal is to construct a
matching in that is popular. A matching is popular if does not lose
a head-to-head election against any matching , where each vertex casts a
vote for the matching in 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 is easy to solve for odd .
Surprisingly, the problem becomes NP-hard for even , as we show here.Comment: Appeared at FSTTCS 201
Popular matchings with two-sided preferences and one-sided ties
We are given a bipartite graph where each vertex has a
preference list ranking its neighbors: in particular, every ranks its
neighbors in a strict order of preference, whereas the preference lists of may contain ties. A matching is popular if there is no matching
such that the number of vertices that prefer to exceeds the number of
vertices that prefer to~. We show that the problem of deciding whether
admits a popular matching or not is NP-hard. This is the case even when
every either has a strict preference list or puts all its neighbors
into a single tie. In contrast, we show that the problem becomes polynomially
solvable in the case when each puts all its neighbors into a single
tie. That is, all neighbors of are tied in 's list and desires to be
matched to any of them. Our main result is an algorithm (where ) for the popular matching problem in this model. Note that this model
is quite different from the model where vertices in have no preferences and
do not care whether they are matched or not.Comment: A shortened version of this paper has appeared at ICALP 201
A new solution concept for the roommate problem
Abstract The aim of this paper is to propose a new solution concept for the roommate problem with strict preferences. We introduce maximum irreversible matchings and consider almost stable matchings (Abraham et al., 2006) and maximum stable matchings (Tan 1990, 1991b). These solution concepts are all core consistent. We find that almost stable matchings are incompatible with the other two concepts. Hence, to solve the roommate problem we propose matchings that lie at the intersection of the maximum irreversible matchings and maximum stable matchings, which we call Q -stable matchings. We construct an efficient algorithm for computing one element of this set for any roommate problem. We also show that the outcome of our algorithm always belongs to an absorbing set (Inarra et al., 2013)
Computational complexity of -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 -stable if no other matching exists that
is more beneficial to at least out of the agents. The concept
generalizes the recently studied majority stability. We prove that whereas the
verification of -stability for a given matching is polynomial-time solvable
in all three models, the complexity of deciding whether a -stable matching
exists depends on and is characteristic to each model.Comment: SAGT 202