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.

The Hospitals / Residents problem with Couples (HRC) is a generalisation of the classical Hospitals / Residents problem (HR) that is important in practical applications because it models the case where couples submit joint preference lists over pairs of hospitals (h i , h j ). We consider a natural restriction of HRC in which the members of a couple have individual preference lists over hospitals, and the joint preference list of the couple is consistent with these individual lists in a precise sense. We give an appropriate stability definition and show that, in this context, the problem of deciding whether a stable matching exists is NP-complete, even if each resident's preference list has length at most 3 and each hospital has capacity at most 2. However, with respect to classical (Gale-Shapley) stability, we give a linear-time algorithm to find a stable matching or report that none exists, regardless of the preference list lengths or the hospital capacities. Finally, for an alternative formulation of our restriction of HRC, which we call the Hospitals / Residents problem with Sizes (HRS), we give a linear-time algorithm that always finds a stable matching for the case that hospital preference lists are of length at most 2, and where hospital capacities can be arbitrary.

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Abstract. An instance of the popular matching problem (POP-M) consists of a set of applicants and a set of posts. Each applicant has a preference list that strictly ranks a subset of the posts. A matching M of applicants to posts is popular if there is no other matching M ′ such that more applicants prefer M ′ to M than prefer M to M ′ . This paper provides a characterization of the set of popular matchings for an arbitrary POP-M instance in terms of a structure called the switching graph, a directed graph computable in linear time from the preference lists. We show that the switching graph can be exploited to yield efficient algorithms for a range of associated problems, including the counting and enumeration of the set of popular matchings and computing popular matchings that satisfy various additional optimality criteria. Our algorithms for computing such optimal popular matchings improve those described in a recent paper by Kavitha and Nasre [5].

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