Sex-Equal Stable Matchings: Complexity and Exact Algorithms

Abstract

We explore the complexity and exact computation of a variant of the classical stable marriage problem in which we seek matchings that are not only stable, but are also “fair” in a formal sense. In particular, we study the sex-equal stable marriage problem (SESM), in which, roughly speaking, we wish to find a stable matching with the property that the men’s happiness is as close as possible to the women’s happiness. This problem is known to be strongly NP-hard (Kato in Jpn. J. Ind. Appl. Math. 10:1–19, 1993). We specifically consider SESM instances in which the preference lists of the men and/or women are bounded in length by a constant. On the negative side, we show that SESM is NP-hard, even if both the men’s and women’s preference lists are of length at most three, and is not even in the class XP when parameterized by the objective value of the solution. This strengthens the NP-hardness results of Kato (Jpn. J. Ind. Appl. Math. 10:1–19, 1993). On the positive side, we show that our hardness result is “tight” by giving a polynomial-time algorithm for the case in which the preference lists on one side (say the men) are of length at most two, and the lengths of the lists on the other side (the women) are unbounded. Furthermore, we give a low-order exponential-time algorithm for SESM in which the preference lists on one side are of length at most l (and the lengths of the lists on the other side are unbounded). In particular, for every pair of constants $l \in\mathcal{Z}^{+}$ and $\epsilon\in\mathcal{R}^{+}$ there is an algorithm with running time bounded by $O^{\star}(2^{(5 - \sqrt{24})(l-2 + \epsilon)n}) + O^{\star}(2^{\frac {(l-1)}{2\epsilon}})$ . Hence, if ϵ is chosen to be a sufficiently small constant, the running time is in O ⋆(1.0726 n ), O ⋆(1.1504 n ), O ⋆(1.2339 n ),… for l=3,4,5,… .