Popular Matchings of Desired Size

Abstract

Our input is an instance of the stable marriage problem with strict and possibly incomplete lists, i.e., it is a bipartite graph $G = (A \cup B,E)$ where each vertex has a strict preference list ranking its neighbors. We consider a generalization of stable matchings called popular matchings: a matching M in G is popular if there is no matching $M'$ such that the vertices that prefer $M'$ to M outnumber those that prefer M to $M'$.

There are linear time algorithms to compute a min-size popular matching and a max-size popular matching in G. The following question is a natural variant of the min-size and max-size popular matching problems:

• given a parameter k, is there a popular matching of size k in G?

Here $\mathsf {min}< k < \mathsf {max}$, where $\mathsf {min}$ and $\mathsf {max}$ are the sizes of a min-size and a max-size popular matching in G. We show the above problem is $\mathsf {NP}$-hard. For any $\mathsf {min}< k < \mathsf {max}$, we also show a linear time algorithm to construct a matching of size k whose unpopularity factor is at most 2.