Counting Houses of Pareto Optimal Matchings in the House Allocation Problem

Abstract

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 every house b ∈ B. We call an injective mapping τ from A to B a matching. A blocking coalition of τ is a 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 an 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

$$|E^*| \leq \sum_{i = 1,\ldots, m}{\left\lfloor\frac{m}{i}\right\rfloor} = \Theta(m\log m).$$

This is asymptotically tight. A set E ⊆ B is reachable (respectively exactly reachable) if there exists a Pareto optimal matching τ whose image contains E as a subset (respectively equals E). We give bounds for the number of exactly reachable sets.We find that our results hold in the more general setting of multi-matchings, when each applicant a of A is matched with ℓ _a elements of B instead of just one. Further, we give complexity results and algorithms for corresponding algorithmic questions. Finally, we characterize unavoidable houses, i.e., houses that are used by all POM’s. This yields efficient algorithms to determine all unavoidable elements.