Instances of small size with no weakly stable matching for three-sided problem with complete cyclic preferences

TL;DR

This paper constructs a counterexample of size 20 for the three-sided stable matching problem with cyclic preferences, disproving a prior conjecture that solutions always exist for such problems.

Contribution

It reduces the known size of a counterexample for the 3D cyclic stable matching problem from 90 to 20, providing new insights into the problem's complexity.

Findings

Counterexample of size 20 disproves the conjecture

The problem of always finding a stable matching does not hold for all instances

Discussion of a variant where solutions always exist

Abstract

Given $n$ men, $n$ women, and $n$ dogs, we assume that each man has a complete preference list of women, while each woman does a complete preference list of dogs, and each dog does a complete preference list of men. We study the so-called 3D-CYC problem, i.e., a three-dimensional problem with cyclic preferences. We understand a matching as a collection of $n$ nonintersecting triples, each of which contains a man, a woman, and a dog. A matching is said to be nonstable, if one can find a man, a woman, and a dog, which belong to different triples and prefer each other to their current partners in the corresponding triples. Otherwise, the matching is said to be stable. According to the conjecture proposed by Eriksson, S\"ostrand, and Strimling (2006), the problem of finding a stable matching (the problem 3DSM-CYC) always has a solution. However, Lam and Paxton have proposed an algorithm for constructing preference lists in 3DSM-CYC of size $n=90$, which has allowed them to disprove the mentioned conjecture. The question on the existence of counterexamples of a lesser size remained open. The main value of this paper consists in reducing the size of the counterexample to $n=20$. At the end part of the paper, we discuss a new variant of 3DSM, whose solution always exists.