Approximation algorithms for the sex-equal stable marriage problem

Abstract

The stable marriage problem is a classical matching problem introduced by Gale and Shapley. It is known that for any instance, there exists a solution, and there is a polynomial time algorithm to find one. However, the matching obtained by this algorithm is man-optimal, that is, the matching is preferable for men but unpreferable for women, (or, if we exchange the role of men and women, the resulting matching is woman-optimal). The sex-equal stable marriage problem posed by Gus-field and Irving asks to find a stable matching "fair" for both genders, namely it asks to find a stable matching with the property that the sum of the men's score is as close as possible to that of the women's. This problem is known to be strongly NP-hard. In this paper, we give a polynomial time algorithm for finding a near optimal solution in the sex-equal stable marriage problem. Furthermore, we consider the problem of optimizing additional criterion: among stable matchings that are near optimal in terms of the sex-equality, find a minimum egalitarian stable matching. We show that this problem is NP-hard, and give a polynomial time algorithm whose approximation ratio is less than two. © Springer-Verlag Berlin Heidelberg 2007.