On the complexity of distributed stable matching with small messages

We consider the distributed complexity of the stable matching problem (a.k.a. "stable marriage"). In this problem, the communication graph is undirected and bipartite, and each node ranks its neighbors. Given a matching of the nodes, a pair of unmatched nodes is called blocking if they prefer each other to their assigned match. A matching is called stable if it does not induce any blocking pair. In the distributed model, nodes exchange messages in each round over the communication links, until they find a stable matching. We show that if messages may contain at most B bits each, then any distributed algorithm that solves the stable matching problem requires Ω√n/B\log n}) communication rounds in the worst case, even for graphs of diameter O(log n), where n is the number of nodes in the graph. Furthermore, the lower bound holds even if we allow the output to contain {O(√n)} blocking pairs, and if a pair is considered blocking only if they like each other much more then their assigned match.