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.
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Huang, CC., Kavitha, T. (2011). Near-Popular Matchings in the Roommates Problem. In: Demetrescu, C., Halldórsson, M.M. (eds) Algorithms – ESA 2011. ESA 2011. Lecture Notes in Computer Science, vol 6942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23719-5_15
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DOI: https://doi.org/10.1007/978-3-642-23719-5_15
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