Popular Matchings with Variable Job Capacities

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We consider the problem of matching people to jobs, where each person ranks a subset of jobs in an order of preference, possibly involving ties. There are several notions of optimality about how to best match each person to a job; in particular, popularity is a natural and appealing notion of optimality. However, popular matchings do not always provide an answer to the problem of determining an optimal matching since there are simple instances that do not admit popular matchings. This motivates the following extension of the popular matchings problem:

  • Given a graph \(G = ({\mathcal{A}}\cup{\mathcal{J}},E)\) where \({\mathcal{A}}\) is the set of people and \({\mathcal{J}}\) is the set of jobs, and a list \(\langle c_1,\ldots,c_{|{\mathcal{J}}|}\rangle\) denoting upper bounds on the capacities of each job, does there exist \((x_1,\ldots,x_{|{\mathcal{J}}|})\) such that setting the capacity of i-th job to x i , where 1 ≤ x i  ≤ c i , for each i, enables the resulting graph to admit a popular matching.

In this paper we show that the above problem is NP-hard. We show that the problem is NP-hard even when each c i is 1 or 2.