A Note on Distributed Stable Matching
Alex Kipnis, Boaz Patt-Shamir · 2009
We consider the distributed complexity of the stable marriage problem. 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 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 marriage problem requires Omega(sqrt(n/(B log n))) communication rounds in the worst case, even for graphs of diameter Theta (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(sqrt(n)) blocking pairs. We also consider epsilon-stability, where a pair is called epsilon-blocking if they can improve the quality of their match by more than an epsilon fraction, for some 0