Multi-Dimensional Stable Roommates in 2-Dimensional Euclidean Space

We investigate the Euclidean d-Dimensional Stable Roommates problem, which asks whether a given set V of d n points from the 2-dimensional Euclidean space can be partitioned into n disjoint (unordered) subsets Π = {V1, , Vn} with |Vi| = d for each Vi ϵ Π such that Π is stable. Here, stability means that no point subset W ⊆ V is blocking Π, and W is said to be blocking Πif |W| = d such that Σ w ϵW δ(w,w) < Σ vϵΠ(w) δ(w, v) holds for each point w ϵ W, where Π (w) denotes the subset Vi ϵ Π which contains w and δ(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 ≥ 3 unless P=NP. Our result for d = 3 answers a decade-long open question in the theory of Stable Matching and Hedonic Games [18, 1, 10, 26, 21].