Near-Popular Matchings in the Roommates Problem
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 Match Satisfying Assignment Preference List Arbitrary Permutation Stable Partition
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