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Title: Near-Popular Matchings in the Roommates Problem
Dr. Chien-Chung Huang, Max Planck Institute
Date & Time: September 29, 2011 14:15
Venue: Conference Room, 01st floor, C Block, Dept. of CSE, Kanwal Rekhi Bldg.
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 {\em 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 {\em 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 $0(\log|V|)$ and such a matching can be computed in linear time. In our problem the optimal matching would be a {\em 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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