Multi-Dimensional Stable Roommates in 2-Dimensional Euclidean Space

Abstract

We establish NP-completeness for the Euclidean 3-Dimensional Stable Roommates problem, which asks whether a given set V of 3n points from the Euclidean space can be partitioned into n disjoint (unordered) triples Π = {V 1 , . . . , V n } such that Π is stable. Here, stability means that no three points x, y, z ∈ V are blocking Π, and x, y, z ∈ V are said to be blocking Π if the following is satisfied:-δ(x, y) + δ(x, z) < δ(x, x 1 ) + δ(x, x 2 ), -δ(y, x) + δ(y, z) < δ(y, y 1 ) + δ(y, y 2 ), and -δ(z, x) + δ(z, y) < δ(z, z 1 ) + δ(z, z 2 ), where {x, x 1 , x 2 }, {y, y 1 , y 2 }, {z, z 1 , z 2 } ∈ Π, and δ(a, b) denotes the Euclidean distance between a and b.