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The Stable Roommates problem involves matching a set of agents into pairsbased on the agents' strict ordinal preference lists. The matching must bestable, meaning that no two agents strictly prefer each other to their assignedpartners. A number of three-dimensional variants exist, in which agents areinstead matched into triples. Both the original problem and these variants canalso be viewed as hedonic games. We formalise a three-dimensional variant usinggeneral additively separable preferences, in which each agent provides aninteger valuation of every other agent. In this variant, we show that a stablematching may not exist and that the related decision problem is NP-complete,even when the valuations are binary. In contrast, we show that if thevaluations are binary and symmetric then a stable matching must exist and canbe found in polynomial time. We also consider the related problem of finding astable matching with maximum utilitarian welfare when valuations are binary andsymmetric. We show that this optimisation problem is NP-hard and present anovel 2-approximation algorithm.

The Three-Dimensional Stable Roommates Problem with Additively Separable Preferences