Three-dimensional matching instances are rich in stable matchings

Abstract

Extensive studies have been carried out on the Stable Matching problem, but they mostly consider cases where the agents to match belong to either one or two sets. Little work has been done on the three-set extension, despite the many applications in which three-dimensional stable matching (3DSM) can be used. In this paper we study the Cyclic 3DSM problem, a variant of 3DSM where agents in each set only rank the agents from one other set, in a cyclical manner. The question of whether every Cyclic 3DSM instance admits a stable matching has remained open for many years. We give the exact number of stable matchings for the class of Cyclic 3DSM instances where all agents in the same set share the same master preference list. This number is exponential in the size of the instances. We also show through empirical experiments that this particular class contains the most constrained Cyclic 3DSM instances, the ones with the fewest stable matchings. This would suggest that not only do all Cyclic 3DSM instances have at least one stable matching, but they each have an exponential number of them.