Stable marriage with general preferences

TL;DR

This paper generalizes the stable marriage problem to arbitrary preferences, proving NP-completeness for existence in general, providing polynomial algorithms for specific cases, and addressing the open problem of 3D stable matchings.

Contribution

It introduces a broad preference model, establishes computational complexity results, and offers algorithms and formulations for special cases, advancing understanding of stable matchings.

Findings

Deciding stable matchings is NP-complete in the general model.

Polynomial-time algorithm exists for asymmetric preferences.

Deciding 3D stable matchings is NP-complete.

Abstract

We propose a generalization of the classical stable marriage problem. In our model, the preferences on one side of the partition are given in terms of arbitrary binary relations, which need not be transitive nor acyclic. This generalization is practically well-motivated, and as we show, encompasses the well studied hard variant of stable marriage where preferences are allowed to have ties and to be incomplete. As a result, we prove that deciding the existence of a stable matching in our model is NP-complete. Complementing this negative result we present a polynomial-time algorithm for the above decision problem in a significant class of instances where the preferences are asymmetric. We also present a linear programming formulation whose feasibility fully characterizes the existence of stable matchings in this special case. Finally, we use our model to study a long standing open problem regarding the existence of cyclic 3D stable matchings. In particular, we prove that the problem of deciding whether a fixed 2D perfect matching can be extended to a 3D stable matching is NP-complete, showing this way that a natural attempt to resolve the existence (or not) of 3D stable matchings is bound to fail.