eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-09-01
36:1
36:16
10.4230/LIPIcs.ESA.2022.36
article
Multi-Dimensional Stable Roommates in 2-Dimensional Euclidean Space
Chen, Jiehua
1
Roy, Sanjukta
2
1
TU Wien, Austria
Faculty of Information Technology, Czech Technical University in Prague, Czech Republic
We investigate the Euclidean π½-Dimensional Stable Roommates problem, which asks whether a given set V of π½β
n points from the 2-dimensional Euclidean space can be partitioned into n disjoint (unordered) subsets Ξ = {Vβ,β¦,V_{n}} with |V_i| = π½ for each V_i β Ξ such that Ξ is {stable}. Here, {stability} means that no point subset W β V is blocking Ξ , and W is said to be {blocking} Ξ if |W| = π½ such that β_{w' β W}Ξ΄(w,w') < β_{v β Ξ (w)}Ξ΄(w,v) holds for each point w β W, where Ξ (w) denotes the subset V_i β Ξ which contains w and Ξ΄(a,b) denotes the Euclidean distance between points a and b. Complementing the existing known polynomial-time result for π½ = 2, we show that such polynomial-time algorithms cannot exist for any fixed number π½ β₯ 3 unless P=NP. Our result for π½ = 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].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol244-esa2022/LIPIcs.ESA.2022.36/LIPIcs.ESA.2022.36.pdf
stable matchings
multidimensional stable roommates
Euclidean preferences
coalition formation games
stable cores
NP-hardness