Popularity on the 3D-Euclidean Stable Roommates

Abstract: 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.