Minimal instances with no weakly stable matching for three-sided problem with cyclic incomplete preferences

Abstract

Given $n$ men, $n$ women, and $n$ dogs, each man has an incomplete preference list of women, each woman does an incomplete preference list of dogs, and each dog does an incomplete preference list of men. We understand a family as a triple consisting of one man, one woman, and one dog such that each of them enters in the preference list of the corresponding agent. We do a matching as a collection of nonintersecting families (some agents, possibly, remain single). A matching is said to be nonstable, if one can find a man, a woman, and a dog which do not live together currently but each of them would become "happier" if they do. Otherwise the matching is said to be stable (a weakly stable matching in 3-DSMI-CYC problem). We give an example of this problem for $n=3$ where no stable matching exists. Moreover, we prove the absence of such an example for $n<3$. Such an example was known earlier only for $n=6$ (Biro, McDermid, 2010). The constructed examples also allows one to decrease (in two times) the size of the recently constructed analogous example for complete preference lists (Lam, Plaxton, 2019).