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 critical matching in the one-to-one setting [Kavitha T., FSTTCS 2021]. 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.

翻译：我们考虑了在双边偏好和双边下限配额存在下的多对多二分匹配问题。问题的输入是一个二分图G=(A U B, E)，其中A U B中的每个顶点指定了对其邻居的严格偏好序。每个顶点具有一个上限配额和一个下限配额，分别表示从其邻居中可分配的最大和最小顶点数。在具有双边下限配额的多对多设置中，通俗地说，临界匹配是指尽可能满足顶点下限配额的匹配。这是一对一设置中临界匹配定义的自然推广[Kavitha T., FSTTCS 2021]。我们在这给定问题中的目标是寻找临界匹配集中的一个流行匹配。如果在给定匹配集中，一个匹配在与该集合中任何匹配的正面交锋选举中不败，则该匹配是流行的。这里，顶点在匹配对之间进行投票。我们证明临界匹配集中始终存在一个流行匹配。我们提出了一种高效算法来计算这种最大规模的匹配。我们利用对偶证书证明了该匹配的流行性。