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Bounded unpopularity matchings. (English)
Algorithmica 61, No. 3, 738-757 (2011).
Summary: 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 popularity. A matching $M$ is popular if there is no matching $M^{\prime}$ such that more people prefer $M^{\prime}$ to $M$ than the other way around. Determining whether a given instance admits a popular matching and, if so, finding one, was studied by {\it D. J. Abraham} et al. [SIAM J. Comput. 37, No. 4, 1030‒1045 (2007; Zbl 1154.91033)]. If there is no popular matching, a reasonable substitute is a matching whose unpopularity is bounded. We consider two measures of unpopularity ‒ unpopularity factor denoted by $u(M)$ and 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 $u(M)\geq 2$ 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: $H=H _{2},H _{3},\dots ,H _{k }$ 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 $u(M)\leq k - 1$ and $g(M)\le n(1-\frac{2}{k})$. Simulation results suggest that our algorithm finds a matching with low unpopularity in random instances.