Popular matchings with weighted voters
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In the Popular Matching problem, we are given a bipartite graph G = (A ∪ B, E) and for each vertex v∈ A∪ B, strict preferences over the neighbors of v. Given two matchings M and M', matching M is more popular than M' if the number of vertices preferring M to M' is larger than the number of vertices preferring M' to M. A matching M is called popular if there is no matching M' that is more popular than M. We consider a natural generalization of Popular Matching where every vertex has a weight. Then, we call a matching M more popular than matching M' if the weight of vertices preferring M to M' is larger than the weight of vertices preferring M' to M. 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 c > 3 and all vertices on the other side have weight 1.
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