Counting Houses of Pareto Optimal Matchings in the House Allocation Problem

In an instance of the house allocation problem, two sets A and B are given. The set A is referred to as applicants and the set B is referred to as houses. We denote by m and n the size of A and B respectively. In the house allocation problem, we assume that every applicant a ∈ A has a preference list over the set of houses B. We call an injective mapping τ from A to B a matching. A blocking coalition of τ is a non-empty subset A′ of A such that there exists a matching τ ′ that differs from τ only on elements of A′, and every element of A′ improves in τ ′, compared to τ , according to its preference list. If there exists no blocking coalition, we call the matching τ a Pareto optimal matching (POM). A house b ∈ B is reachable if there exists a Pareto optimal matching using b. The set of all reachable houses is denoted by E∗. We show