Approximation Algorithms for Stable Marriage Problems

Approximation Algorithms for Stable Marriage Problems Hiroki Yanagisawa Kyoto University, Japan 2007 The stable marriage problem is a classical matching problem introduced by Gale and Shapley. An instance of the stable marriage problem consists of men and women, where each person totally orders all members of the opposite sex. A matching is stable if there is no pair that prefer each other to their current partners. The problem is to find a stable matching for a given instance. It is known for any instance, there exists a solution, and there is a polynomial time algorithm (Gale-Shapley algorithm) to find one. In Chapter 3, we consider the quality of solutions for the stable marriage problem. The matching obtained by Gale-Shapley 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 sexequal stable marriage problem posed by Gusfield 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. 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. In Chapter 4 and Chapter 5, we consider general settings of the original stable marriage problem. While the original stable marriage problem requires all participants to rank all members of the opposite sex in a strict order, two natural variations are to allow for incomplete preference lists and ties in the preferences. Either variation is polynomially solvable, but it was shown to be NP-hard to find a maximum cardinality stable matching when both of the variations are allowed. It is easy to see that in the generalized variant, the size of any two stable matchings differ by at most a factor of two, and so, an approximation algorithm with a factor two is trivial. In Chapter 4, we give the first nontrivial result for approximation of factor less than two. Our algorithm achieves an approximation ratio of 2/(1+L−2) for instances in which only men have ties of length at most L. When both men and women are allowed to have ties, but the lengths are limited to two, we show a ratio of 13/7 (< 1.858). We also improve the lower bound on the approximation ratio to 33/29 (> 1.1379). In Chapter 5, we give a randomized approximation algorithm and show that its expected approximation ratio is at most 10/7 (< 1.4286) for a restricted but still NP-hard case, where ties occur in only men’s lists, each man writes at most one tie, and the length of ties is two. We also show that our analysis is nearly tight by giving a lower bound 32/23 (> 1.3913) for it. Furthermore, we show that these restrictions except for the last one can be removed without increasing the approximation ratio too much.