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URN: urn:nbn:de:0030-drops-169741
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Multi-Dimensional Stable Roommates in 2-Dimensional Euclidean Space

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Abstract

We investigate the Euclidean š–½-Dimensional Stable Roommates problem, which asks whether a given set V of š–½ā‹… n points from the 2-dimensional Euclidean space can be partitioned into n disjoint (unordered) subsets Ī  = {Vā‚,ā€¦,V_{n}} with |V_i| = š–½ for each V_i āˆˆ Ī  such that Ī  is {stable}. Here, {stability} means that no point subset W āŠ† V is blocking Ī , and W is said to be {blocking} Ī  if |W| = š–½ such that āˆ‘_{w' āˆˆ W}Ī“(w,w') < āˆ‘_{v āˆˆ Ī (w)}Ī“(w,v) holds for each point w āˆˆ W, where Ī (w) denotes the subset V_i āˆˆ Ī  which contains w and Ī“(a,b) denotes the Euclidean distance between points a and b. Complementing the existing known polynomial-time result for š–½ = 2, we show that such polynomial-time algorithms cannot exist for any fixed number š–½ ā‰„ 3 unless P=NP. Our result for š–½ = 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].

BibTeX - Entry

@InProceedings{chen_et_al:LIPIcs.ESA.2022.36,
  author =	{Chen, Jiehua and Roy, Sanjukta},
  title =	{{Multi-Dimensional Stable Roommates in 2-Dimensional Euclidean Space}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{36:1--36:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16974},
  URN =		{urn:nbn:de:0030-drops-169741},
  doi =		{10.4230/LIPIcs.ESA.2022.36},
  annote =	{Keywords: stable matchings, multidimensional stable roommates, Euclidean preferences, coalition formation games, stable cores, NP-hardness}
}

Keywords: stable matchings, multidimensional stable roommates, Euclidean preferences, coalition formation games, stable cores, NP-hardness
Seminar: 30th Annual European Symposium on Algorithms (ESA 2022)
Issue date: 2022
Date of publication: 01.09.2022


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