A necessary and sufficient condition for the existence of a complete stable matching
Journal of Algorithms
EC '06 Proceedings of the 7th ACM conference on Electronic commerce
Popular matchings in the capacitated house allocation problem
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
SIAM Journal on Computing
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Popular matchings in the stable marriage problem
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Popular matchings in the marriage and roommates problems
CIAC'10 Proceedings of the 7th international conference on Algorithms and Complexity
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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.