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 $\epsilon > 0$, it is NP-hard to compute a matching whose unpopularity factor is at most $4/3 - \epsilon$ of the optimal.
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