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

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 wheth...

• Type: Conference
• Year: 2007
• Where: ESA
• Authors: Chien-Chung Huang
• Keywords: ESA 2007, Open Questions, Stable Family Problem, Stable Matching