Popularity at Minimum Cost

Abstract

AbstractWe 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 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 \(b \in \mathcal{B}\) 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. However, when the number of copies of each item is fixed, the problem of computing a minimum cost popular matching or deciding that no popular matching exists can be solved in O(mn 1) time.KeywordsInput GraphExtra CopyMaximum MatchSatisfying AssignmentPreference ListThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.