The Three-Dimensional Stable Roommates Problem with Additively Separable Preferences

Abstract

The Stable Roommates problem involves matching a set of agents into pairs based on the agents' strict ordinal preference lists. The matching must be stable, meaning that no two agents strictly prefer each other to their assigned partners. A number of three-dimensional variants exist, in which agents are instead matched into triples. Both the original problem and these variants can also be viewed as hedonic games. We formalise a three-dimensional variant using general additively separable preferences, in which each agent provides an integer valuation of every other agent. In this variant, we show that a stable matching may not exist and that the related decision problem is NP-complete, even when the valuations are binary. In contrast, we show that if the valuations are binary and symmetric then a stable matching must exist and can be found in polynomial time. We also consider the related problem of finding a stable matching with maximum utilitarian welfare when valuations are binary and symmetric. We show that this optimisation problem is NPhard and present a novel 2-approximation algorithm.

Partitioning Friends Fairly

“…In this case, even with symmetric additive preferences, a solution in the core may not exist (Arkin et al 2009), and checking whether it does is NP-hard (Chen and Roy 2021). However, if we further restrict the preferences to be binary, then McKay and Manlove (2021) show that a solution in the core always exists and can be found efficiently. Our problem can be seen as a multidimensional generalization of the roommate problem with symmetric binary additive preferences.…”