Characterizing a Set of Popular Matchings Defined by Preference Lists with Ties

In this paper, we give a characterization of a set of popular matchings in a bipartite graph with one-sided preference lists. The concept of a popular matching was first introduced by Gardenfors [5]. Recently, Abraham et al. [1] discussed a problem for finding a popular matching and proposed polynomial time algorithms for problem instances defined by preference lists with or without ties. McDermid and Irving [10] discussed a set of popular matchings defined by strict preference lists. One of a remained open problems raised in [8, 10] and [9] (Section 7.7) is a characterization of a set of popular matchings when given preference lists have ties. This paper solves the above open problem affirmatively and gives an explicit characterization of a set of popular matchings defined by preference lists with ties. By employing our characterization, we can transform a minimum cost popular matching problem, discussed in [8, 10], to a simple minimum cost assignment problem.