Abstract
In this paper, we consider a matching in a bipartite graph where each vertex has a preference over the edges incident to it. Especially, we consider the case where a preference of a vertex is defined based on comparisons of any two edges incident to it. In this setting, we first consider the problem of checking the existence of a Pareto efficient matching, and we prove that this problem is NP-complete even in some restricted situation. Furthermore, we consider the problem of checking the existence of a Pareto stable matching, which is a matching satisfying stability and Pareto efficiency, under the above general preferences. We first prove that this problem is NP-complete even when there exist a stable matching and a Pareto efficient matching. Then we propose a polynomial-time algorithm for the problem of checking the existence of a Pareto stable matching in trees.
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