The Three-Dimensional Stable Roommates Problem with Additively Separable Preferences

TL;DR

This paper studies a three-dimensional stable roommates problem with additively separable preferences, revealing conditions for existence, computational complexity, and approximation algorithms for maximizing welfare.

Contribution

It formalizes a new three-dimensional variant with additively separable preferences, analyzes its computational complexity, and provides algorithms for stable matchings and welfare maximization.

Findings

Stable matchings may not always exist.

Deciding stability is NP-complete with binary valuations.

Stable matchings always exist and are polynomial-time computable with symmetric binary valuations.

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