## 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 *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 by Abraham et al. (SIAM J. Comput. 37(4):1030–1045, 2007). 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*)≥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},…,*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*)≤*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.

### Keywords

Matching with preferences Popularity Approximation algorithms## Preview

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