74 research outputs found
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 weighted voters
In the Popular Matching problem, we are given a bipartite graph and for each vertex , strict preferences over the
neighbors of . Given two matchings and , matching is more
popular than if the number of vertices preferring to is larger
than the number of vertices preferring to . A matching is called
popular if there is no matching that is more popular than .
We consider a natural generalization of Popular Matching where every vertex
has a weight. Then, we call a matching more popular than matching if
the weight of vertices preferring to is larger than the weight of
vertices preferring to . For this case, we show that it is NP-hard to
find a popular matching. Our main result its a polynomial-time algorithm that
delivers a popular matching or a proof for it non-existence in instances where
all vertices on one side have weight and all vertices on the other side
have weight 1
Matchings with lower quotas: Algorithms and complexity
We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G=(A∪˙P,E)G=(A∪˙P,E) with weights on the edges in E, and with lower and upper quotas on the vertices in P. We seek a maximum weight many-to-one matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-to-one matching with lower and upper quotas (WMLQ), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of WMLQ from the viewpoints of classical polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between NPNP-hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota umaxumax as basis, and we prove that this dependence is necessary unless FPT=W[1]FPT=W[1]. The approximability of WMLQ is also discussed: we present an approximation algorithm for the general case with performance guarantee umax+1umax+1, which is asymptotically best possible unless P=NPP=NP. Finally, we elaborate on how most of our positive results carry over to matchings in arbitrary graphs with lower quotas
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
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