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 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 showThis 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 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 POMs. We obtain efficient algorithms to determine all unavoidable elements.

Set systems related to a house allocation problem

“…Using the result of [2] this implied for them the upper bound O(n 2 m 3.5 (e ln m + e) m ) for the complexity of D B,A . Our Theorem 1.5 improves this to O(n 2 m 3.5 4 m ).…”

“…For any matching τ the set {τ (a) : a ∈ A} of houses which are taken is denoted by s(τ ). The following lemma from [2] summarizes all that we will use in this paper about POMs. We say that a subset E of B is reachable if there exists a POM τ with s(τ ) = E. Lemma 1.1 (Lemma 7 in [2]).…”

“…The following lemma from [2] summarizes all that we will use in this paper about POMs. We say that a subset E of B is reachable if there exists a POM τ with s(τ ) = E. Lemma 1.1 (Lemma 7 in [2]). Let E ⊆ {1, ..., n} with |E| = m. The following statements are equivalent.…”

“…Therefore, g(m) ≤ h(m). The following lemma is hidden in the proof of the upper bound on the number of reachable houses from [2]. Lemma 1.4 ([2]).…”

Minimum Partial-Matching and Hausdorff RMS-Distance under Translation: Combinatorics and Algorithms

“…Lemma 6 (Asinowski et al [3]) The number of elements that belong to some efficient matching with respect to m ordered preference lists is at most m(ln m + 1).…”