Abstract
Our input is a graph G = (V, E) where each vertex ranks its neighbors in a strict order of preference. The problem is to compute a matching in G that captures the preferences of the vertices in a popular way. Matching M is more popular than matching M′ if the number of vertices that prefer M to M′ is more than those that prefer M′ to M. The unpopularity factor of M measures by what factor any matching can be more popular than M. We show that G always admits a matching whose unpopularity factor is O(log|V|) and such a matching can be computed in linear time. In our problem the optimal matching would be a least unpopularity factor matching - we show that computing such a matching is NP-hard. In fact, for any ε > 0, it is NP-hard to compute a matching whose unpopularity factor is at most 4/3 − ε of the optimal.KeywordsStable MatchSatisfying AssignmentPreference ListArbitrary PermutationStable PartitionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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