Solving hard stable matching problems involving groups of similar agents

Many important stable matching problems are known to be NP-hard, even when strong restrictions are placed on the input. In this paper we seek to identify simple structural properties of instances of stable matching problems which will allow the design of efficient algorithms. We focus on the setting in which all agents involved in some matching problem can be partitioned into k different types, where the type of an agent determines his or her preferences, and agents have preferences over types (which may be refined by more detailed preferences within a single type). This situation could arise in practice if agents form preferences based on some small collection of agents' attributes. The notion of types could also be used if we are interested in a relaxation of stability, in which agents will only form a private arrangement if it allows them to be matched with a partner who differs from the current partner in some particularly important characteristic. We show that in this setting several well-studied NP-hard stable matching problems (such as MAX SMTI, MAX SRTI, and MAX SIZE MIN BP SMTI) belong to the parameterised complexity class FPT when parameterised by the number of different types of agents, and so admit efficient algorithms when this number of types is small