Multi-Dimensional Stable Roommates in 2-Dimensional Euclidean Space

We investigate the Euclidean $d$-Dimensional Stable Roommates problem, which asks whether a given set~$V$ of $d \cdot n$ points from the 2-dimensional Euclidean space can be partitioned into $n$ disjoint (unordered) subsets $\Pi=\{V_1,\ldots,V_{n}\}$ with $|V_i|=d$ for each $V_i\in \Pi$ such that $\Pi$ is stable. Here, stability means that no point subset $W\subseteq V$ is blocking $\Pi$ and $W$ is said to be blocking $\Pi$ if $|W|= d$ such that $\sum_{w'\in W}\delta(w,w') < \sum_{v\in \Pi(w)}\delta(w,v)$ holds for each point $w\in W$, where $\Pi(w)$ denotes the subset $V_i\in \Pi$ which contains $w$ and $\delta(a,b)$ denotes the Euclidean distance between points $a$ and $b$. Complementing the existing known polynomial-time result for $d=2$, we show that such polynomial-time algorithms cannot exist for any fixed number $d \ge 3$ unless P=NP. Our result for $d=3$ answers a decade-long open question in the theory of Stable Matching and Hedonic Games [17, 1, 9, 25, 20].