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Three-Sided Stable Matchings with Cyclic Preferences

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Abstract

Knuth (Mariages Stables, Les Presses de L’Université de Montréal, 1976) asked whether the stable matching problem can be generalised to three dimensions, e.g., for families containing a man, a woman and a dog. Subsequently, several authors considered the three-sided stable matching problem with cyclic preferences, where men care only about women, women only about dogs, and dogs only about men. In this paper we prove that if the preference lists may be incomplete, then the problem of deciding whether a stable matching exists, given an instance of the three-sided stable matching problem with cyclic preferences, is NP-complete. Considering an alternative stability criterion, strong stability, we show that the problem is NP-complete even for complete lists. These problems can be regarded as special types of stable exchange problems, therefore these results have relevance in some real applications, such as kidney exchange programs.

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Authors and Affiliations

  1. Department of Computing Science, University of Glasgow, Glasgow, G12 8QQ, UK

    Péter Biró & Eric McDermid

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  1. Péter Biró
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  2. Eric McDermid
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Correspondence to Péter Biró.

Additional information

This work was supported by EPSRC grant EP/E011993/1.

P. Biró was supported by OTKA grant K69027.

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Biró, P., McDermid, E. Three-Sided Stable Matchings with Cyclic Preferences. Algorithmica 58, 5–18 (2010). https://doi.org/10.1007/s00453-009-9315-2

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  • DOI: https://doi.org/10.1007/s00453-009-9315-2

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