Multi-Dimensional Stable Roommates in 2-Dimensional Euclidean Space
We investigate the Euclidean đ˝-Dimensional Stable Roommates problem, which asks whether a given set V of đ˝â
n points from the 2-dimensional Euclidean space can be partitioned into n disjoint (unordered) subsets Î = {Vâ,âŚ,V_{n}} with |V_i| = đ˝ for each V_i â Î such that Î is {stable}. Here, {stability} means that no point subset W â V is blocking Î , and W is said to be {blocking} Î if |W| = đ˝ such that â_{w' â W}Î´(w,w') < â_{v â Î (w)}Î´(w,v) holds for each point w â W, where Î (w) denotes the subset V_i â Î which contains w and Î´(a,b) denotes the Euclidean distance between points a and b. Complementing the existing known polynomial-time result for đ˝ = 2, we show that such polynomial-time algorithms cannot exist for any fixed number đ˝ âĽ 3 unless P=NP. Our result for đ˝ = 3 answers a decade-long open question in the theory of Stable Matching and Hedonic Games [Iwama et al., 2007; Arkin et al., 2009; Vladimir G. Deineko and Gerhard J. Woeginger, 2013; Vladimir G. Deineko and Gerhard J. Woeginger, 2013; David F. Manlove, 2013].
stable matchings
multidimensional stable roommates
Euclidean preferences
coalition formation games
stable cores
NP-hardness
Theory of computation~Problems, reductions and completeness
Theory of computation~Solution concepts in game theory
Theory of computation~Computational geometry
36:1-36:16
Regular Paper
https://arxiv.org/abs/2108.03868
Jiehua
Chen
Jiehua Chen
TU Wien, Austria
Vienna Science and Technology Fund (WWTF) grant VRG18-012.
Sanjukta
Roy
Sanjukta Roy
Faculty of Information Technology, Czech Technical University in Prague, Czech Republic
TU Wien, Austria
This work was done when SR was affiliated with TU Vienna, and was supported by Vienna Science and Technology Fund (WWTF) grant VRG18-012.
10.4230/LIPIcs.ESA.2022.36
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Jiehua Chen and Sanjukta Roy
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