We study the 3D-Euclidean Multidimensional Stable Roommates problem, which asks whether a given set $V$ of $s\cdot n$ agents with a location in 3-dimensional Euclidean space can be partitioned into $n$ disjoint subsets $\pi = \{R_1 ,\dots , R_n\}$ with $|R_i| = s$ for each $R_i \in \pi$ such that $\pi$ is (strictly) popular, where $s$ is the room size. A partitioning is popular if there does not exist another partitioning in which more agents are better off than worse off. Computing a popular partition in a stable roommates game is NP-hard, even if the preferences are strict. The preference of an agent solely depends on the distance to its roommates. An agent prefers to be in a room where the sum of the distances to its roommates is small. We show that determining the existence of a strictly popular outcome in a 3D-Euclidean Multidimensional Stable Roommates game with room size $3$ is co-NP-hard.

翻译：我们研究3D-欧几里得多维稳定室友问题，该问题询问：给定一个由$s\cdot n$个智能体组成的集合$V$，每个智能体在三维欧几里得空间中具有一个位置，能否将其划分为$n$个不相交的子集$\pi = \{R_1 ,\dots , R_n\}$，其中每个$R_i \in \pi$满足$|R_i| = s$（$s$为房间大小），使得划分$\pi$是（严格）流行的。一个划分被称为流行的，如果不存在另一个划分使更多智能体的情况变好而非变差。在稳定室友博弈中计算一个流行划分是NP难的，即使偏好是严格的。智能体的偏好完全取决于其与室友的距离，智能体倾向于加入一个使其与室友距离总和较小的房间。我们证明，在房间大小为$3$的3D-欧几里得多维稳定室友博弈中，判断严格流行结果的存在性是co-NP难的。