Pairwise preferences in the stable marriage problem
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.