  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 NP-hard and present a
novel 2-approximation algorithm.
