Three-Sided Stable Matchings with Cyclic Preferences

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.