Two’s Company, Three’s a Crowd: Stable Family and Threesome Roommates Problems

Abstract

We investigate Knuth’s eleventh open question on stable matchings. In the stable family problem, sets of women, men, and dogs are given, all of whom state their preferences among the other two groups. 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. A similar problem, called the threesome roommates problem, assumes that a group of persons, each with their preferences among the combinations of two others, are to be partitioned into triples. Similarly, the goal is to make sure that no three persons want to break up with their assigned roommates.

Ng and Hirschberg were the first to investigate these two problems. In their formulation, each participant provides a strictly-ordered list of all combinations. They proved that under this scheme, both problems are NP-complete. Their paper reviewers pointed out that their reduction exploits inconsistent preference lists and they wonder whether these two problems remain NP-complete if preferences are required to be consistent. We answer in the affirmative.

In order to give these two problems a broader outlook, we also consider the possibility that participants can express indifference, on the condition that the preference consistency has to be maintained. As an example, we propose a scheme in which all participants submit two (or just one in the roommates case) lists ranking the other two groups separately. The order of the combinations is decided by the sum of their ordinal numbers. Combinations are tied when the sums are equal. By introducing indifference, a hierarchy of stabilities can be defined. We prove that all stability definitions lead to NP-completeness for existence of a stable matching.