Popular Critical Matchings in the Many-to-Many Setting

We consider the many-to-many bipartite matching problem in the presence of two-sided preferences and two-sided lower quotas. The input to our problem is a bipartite graph G=(A U B, E), where  each vertex in A U B specifies a strict preference ordering over its neighbors. Each vertex has an upper quota and a lower quota denoting the maximum and minimum number of vertices that can be assigned to it from its neighborhood.In the many-to-many setting with two-sided lower quotas, informally, a critical matching is a matching which fulfils vertex lower quotas to the maximum possible extent. This is a natural generalization of the definition of a critical matching in the one-to-one setting. Our goal in the given problem is to find a popular matching in the set of critical matchings. A matching is popular in a given set of matchings if it remains undefeated in a head-to-head election with any matching in that set. Here, vertices cast votes between pairs of matchings. We show that there always exists a matching that is popular in the set of critical matchings. We present an  efficient algorithm to compute such a matching of the largest size. We prove the popularity of our matching using a dual certificate.