A Matroid Approach to Stable Matchings with Lower Quotas
针对有下限配额的稳定匹配问题，提出基于拟阵的通用解法，证明其多项式时间可解，并给出稳定分配的几何描述和格结构。
In 2010, Huang introduced the laminar classified stable matching problem (lcsm for short) that is motivated by academic hiring. This problem is an extension of the well-known hospitals/residents problem in which a hospital has laminar classes of residents and it sets lower and upper bounds on the number of residents that it can hire in each class. Against the intuition that variations of the stable matching problem with lower quotas are difficult in general, Huang proved that lcsm can be solved in polynomial time. In this paper, we present a matroid-based approach to lcsm and we obtain the following results. (i) We solve a generalization of lcsm in which both sides have quotas. (ii) Huang raised a question about a polyhedral description of the set of stable assignments in lcsm. We give a positive answer for this question by exhibiting a polyhedral description of the set of stable assignments in a generalization of lcsm. (iii) We prove that the set of stable assignments in a generalization of lcsm has a lattice structure that is similar to the (ordinary) stable matching problem.