Euclidean 3D Stable Roommates is NP-hard
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We establish NP-completeness for the Euclidean 3D 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.
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