Dynamic Pareto Optimal Matching

We consider the problem of assigning houses to agents, where each agent has his own partial ranking of the houses (i.e., a preference list of a subset of the houses). A matching M is Pareto optimal if there exists no other matching M´ such that all agents have a house in M´ not worse than in M and at least one agent has a better house in M´. In a dynamic market where agents and houses can be added or removed at any time, a Pareto optimal matching may lose its optimality. In this paper, we give an O(m) time algorithm to maintain a maximum cardinality and Pareto optimal matching, where m is the total size of all preference lists. Furthermore, we show that a Pareto optimal matching can be transformed to any other Pareto optimal matching through a certain constructed graph in linear time.