Multi-Dimensional Stable Roommates in 2-Dimensional Euclidean Space

Abstract

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 $\Pi =\{V_1,\dots ,V_n\}$ 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 ${\sum }_{w^{\prime }\in W}\delta (w,w^{\prime })<{\sum }_{v\in \Pi (w)}\delta (w,v)$ 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].