Near-Popular Matchings in the Roommates Problem


<p>Chien-Chung Huang<br /> Humboldt-UniversitÃ&curren;t zu<br /> Berlin Institut fÃ&frac14;r Informatik 10099<br /> Berlin Germany</p>


Monday, 26 September 2011 (All day)


  • A-212 (STCS Seminar Room)

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 will show that computing such a matching is NP-hard. In fact, for any $\epsilon$, it is NP-hard to compute a matching whose unpopularity factor is at most $4/3 - \epsilon$ of the optimal.