## Computer Science Technical Reports

#### Title

Bounded Unpopularity Matchings

Technical Report

4-1-2008

TR2008-616

#### Abstract

We investigate the following problem: given a set of jobs and a set of people with preferences over the jobs, what is the optimal way of matching people to jobs? Here we consider the notion of \emph{popularity}. A matching $M$ is popular if there is no matching $M'$ such that more people prefer $M'$ to $M$ than the other way around. Determining whether a given instance admits a popular matching and, if so, finding one, was studied in \cite{AIKM05}. If there is no popular matching, a reasonable substitute is a matching whose {\em unpopularity} is bounded. We consider two measures of unpopularity - {\em unpopularity factor} denoted by $u(M)$ and {\em unpopularity margin} denoted by $g(M)$. McCutchen recently showed that computing a matching $M$ with the minimum value of $u(M)$ or $g(M)$ is NP-hard, and that if $G$ does not admit a popular matching, then we have $u(M) \ge 2$ for all matchings $M$ in $G$. Here we show that a matching $M$ that achieves $u(M) = 2$ can be computed in $O(m\sqrt{n})$ time (where $m$ is the number of edges in $G$ and $n$ is the number of nodes) provided a certain graph $H$ admits a matching that matches all people. We also describe a sequence of graphs: $H = H_2, H_3,\ldots,H_k$ such that if $H_k$ admits a matching that matches all people, then we can compute in $O(km\sqrt{n})$ time a matching $M$ such that $u(M) \le k-1$ and $g(M) \le n(1-\frac{2}{k})$. Simulation results suggest that our algorithm finds a matching with low unpopularity.

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