Circular stable matching and 3-way kidney transplant

We consider the following version of the stable matching problem. Suppose that men have preferences for women, women have preferences for dogs, and dogs have preferences for men. The goal is to organize them into family units so that no three of them have incentive to desert their assigned family members to join in a new family. This problem is called circular stable matching, allegedly originated by Knuth. We also investigate a generalized version of this problem, in which every participant has preference among all others. The goal is similarly to partition them into oriented triples so that no three persons have incentive to deviate from the assignment. This problem is motivated by recent innovations in kidney exchange, and we call it the 3-way kidney transplant problem. We report complexity, structural and counting results on these two problems.