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 = (A ∪ B, E), where A is a set of people, B is a set of items, and each person a ∈ A ranks a subset of items in an 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 ∈ 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 √ n1/2, where n1 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 (i.e., its capacity) 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(mn1) time.