11 research outputs found
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
Popular and Dominant Matchings with Uncertain, Multilayer and Aggregated Preferences
We study the Popular Matching problem in multiple models, where the
preferences of the agents in the instance may change or may be
unknown/uncertain. In particular, we study an Uncertainty model, where each
agent has a possible set of preferences, a Multilayer model, where there are
layers of preference profiles, a Robust model, where any agent may move some
other agents up or down some places in his preference list and an Aggregated
Preference model, where votes are summed over multiple instances with different
preferences.
We study both one-sided and two-sided preferences in bipartite graphs. In the
one-sided model, we show that all our problems can be solved in polynomial time
by utilizing the structure of popular matchings. We also obtain nice structural
results. With two-sided preferences, we show that all four above models lead to
NP-hard questions for popular matchings. By utilizing the connection between
dominant matchings and stable matchings, we show that in the robust and
uncertainty model, a certainly dominant matching in all possible prefernce
profiles can be found in polynomial-time, whereas in the multilayer and
aggregated models, the problem remains NP-hard for dominant matchings too.
We also answer an open question about -robust stable matchings
Matchings and Copeland's Method
Given a graph 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 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 .
We introduce a relaxation of both popular matchings and Copeland winners
called weak Copeland winners. These are matchings with Copeland score at least
, where is the total number of matchings in ; the maximum
possible Copeland score is . We show a fully polynomial-time
randomized approximation scheme to compute a matching with Copeland score at
least for any
Popular matchings with two-sided preferences and one-sided ties
We are given a bipartite graph G = (A âȘ B, E) where each vertex has a preference list ranking its neighbors: in particular, every a â A ranks its neighbors in a strict order of preference, whereas the preference lists of b â B may contain ties. A matching M is popular if there is no matching Mâ such that the number of vertices that prefer Mâ to M exceeds the number that prefer M to Mâ. We show that the problem of deciding whether G admits a popular matching or not is NP-hard. This is the case even when every b â B 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 b â B puts all its neighbors into a single tie. That is, all neighbors of b are tied in bâs list and and b desires to be matched to any of them. Our main result is an O(n2) algorithm (where n = |A âȘ B|) for the popular matching problem in this model. Note that this model is quite different from the model where vertices in B have no preferences and do not care whether they are matched or not
Popular matchings with two-sided preferences and one-sided ties
We are given a bipartite graph G = (A âȘ B, E) where each vertex has a preference list ranking its neighbors: in particular, every a â A ranks its neighbors in a strict order of preference, whereas the preference lists of b â B may contain ties. A matching M is popular if there is no matching Mâ such that the number of vertices that prefer Mâ to M exceeds the number that prefer M to Mâ. We show that the problem of deciding whether G admits a popular matching or not is NP-hard. This is the case even when every b â B 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 b â B puts all its neighbors into a single tie. That is, all neighbors of b are tied in bâs list and and b desires to be matched to any of them. Our main result is an O(n2) algorithm (where n = |A âȘ B|) for the popular matching problem in this model. Note that this model is quite different from the model where vertices in B have no preferences and do not care whether they are matched or not