Distributed Weighted Stable Marriage Problem

Abstract

The Stable Matching problem was introduced by Gale and Shapley in 1962. The input for the stable matching problem is a complete bipartite K n,n graph together with a ranking for each node. Its output is a matching that does not contain a blocking pair, where a blocking pair is a pair of elements that are not matched together but rank each other higher than they rank their current mates. In this work we study the Distributed Weighted Stable Matching problem. The input to the Weighted Stable Matching problem is a complete bipartite K n,n graph and a weight function W. The ranking of each node is determined by W, i.e. node v prefers node u 1 over node u 2 if W((v,u 1)) > W((v, u 2)). Using this ranking we can solve the original Stable Matching problem. We consider two different communication models: the billboard model and the full distributed model. In the billboard model, we assume that there is a public billboard and each participant can write one message on it in each time step. In the distributed model, we assume that each node can send O(logn) bits on each edge of the K n,n . In the billboard model we prove a somewhat surprising tight bound: any algorithm that solves the Stable Matching problem requires at least n − 1 rounds. We provide an algorithm that meets this bound. In the distributed communication model we provide an algorithm named intermediation agencies algorithm, in short (IAA), that solves the Distributed Weighted Stable Marriage problem in \(O(\sqrt{n})\) rounds. This is the first sub-linear distributed algorithm that solves some subcase of the general Stable Marriage problem.KeywordsStable MarriageDistributed AlgorithmsMatchingBillboardScheduling