Popularity at minimum cost

We consider an extension of the popular matching problem in this paper. The input to the popular matching problem is a bipartite graph $G = (\mathcal{A}\cup\mathcal{B},E)$ , where $\mathcal{A}$ is a set of people, $\mathcal{B}$ is a set of items, and each person $a \in\mathcal{A}$ ranks a subset of items in order of preference, with ties allowed. The popular matching problem seeks to compute a matching M ∗ between people and items such that there is no matching M where more people are happier with M than with M ∗. Such a matching M ∗ is called a popular matching. However, there are simple instances where no popular matching exists. Here we consider the following natural extension to the above problem: associated with each item $b \in\mathcal{B}$ is a non-negative price cost(b), that is, for any item b, new copies of b can be added to the input graph by paying an amount of cost(b) per copy. When G does not admit a popular matching, the problem is to “augment” G at minimum cost such that the new graph admits a popular matching. We show that this problem is NP-hard; in fact, it is NP-hard to approximate it within a factor of $\sqrt{n_{1}}/2$ , where n 1 is the number of people. This problem has a simple polynomial time algorithm when each person has a preference list of length at most 2. However, if we consider the problem of constructing a graph at minimum cost that admits a popular matching that matches all people, then even with preference lists of length 2, the problem becomes NP-hard. On the other hand, when the number of copies of each item is fixed, we show that the problem of computing a minimum cost popular matching or deciding that no popular matching exists can be solved in O(mn 1) time, where m is the number of edges.