Counterexamples of Small Size for Three-Sided Stable Matching with Cyclic Preferences

Abstract

Abstract: Given n men, n women, and n dogs, let each man have a complete preference list of women, each woman have a complete preference list of dogs, and each dog have a complete preference list of men (three-dimensional problem with cyclic preferences, also known as the so-called 3D-CYC problem). A matching is a collection of n nonintersecting triples, each of which contains one representative of each gender. A matching is called a stable matching (SM) if one cannot find a man, a woman, and a dog belonging to different triples and preferring each other to their current partners in the corresponding triples. Eriksson et al. [2] hypothesized that the 3DSM-CYC problem, which implies the search of SM, always has a stable matching. Gradually the conjecture was proved for all n ≤ 5. However, Lam and Plaxton [1] proposed an algorithm for constructing preference lists for the 3DSM-CYC problem of size n = 90, for which no SM exists. The question on the existence of counterexamples of a lesser size remained open. In this paper, we construct a vivid counterexample of the 3DSM-CYC problem of size n = 24.