Counting One-sided Exchange Stable Matchings

Let A,B with |A| = m and |B| = n ≥ m be two sets. We assume that every element a ∈ A has a preference list over all elements from 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 exchange stable matching (ESM). An element b ∈ B is reachable if there exists an exchange stable matching using b. The set of all reachable elements is denoted by E∗. We show |E| ≤ ∑ i=1,...,m ⌊m i ⌋ = Θ(m logm). This is asymptotically tight. A set E ⊆ B is reachable (respectively exactly reachable) if there exists an exchange stable 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 element 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 elements, i.e., elements of B that are used by all ESM’s. This yields efficient algorithms to determine all unavoidable elements.