Pairwise preferences in the stable marriage problem

TL;DR

This paper explores the complexity of the stable marriage problem under various preference structures, introducing algorithms and complexity results for different levels of preference orderedness and stability notions.

Contribution

It provides a comprehensive analysis of the stable marriage problem with pairwise preferences, establishing complexity boundaries across six preference orderedness levels.

Findings

Polynomial algorithms for certain preference structures

NP-completeness proofs for other cases

Exact complexity boundaries between tractable and intractable cases

Abstract

We study the classical, two-sided stable marriage problem under pairwise preferences. In the most general setting, agents are allowed to express their preferences as comparisons of any two of their edges and they also have the right to declare a draw or even withdraw from such a comparison. This freedom is then gradually restricted as we specify six stages of orderedness in the preferences, ending with the classical case of strictly ordered lists. We study all cases occurring when combining the three known notions of stability---weak, strong and super-stability---under the assumption that each side of the bipartite market obtains one of the six degrees of orderedness. By designing three polynomial algorithms and two NP-completeness proofs we determine the complexity of all cases not yet known, and thus give an exact boundary in terms of preference structure between tractable and intractable cases.