An efficient algorithm for the “optimal” stable marriage

Abstract

In an instance of size n of the stable marriage problem, each of n men and n women ranks the members of the opposite sex in order of preference. A stable matching is a complete matching of men and women such that no man and woman who are not partners both prefer each other to their actual partners under the matching. It is well known [2] that at least one stable matching exists for every stable marriage instance. However, the classical Gale-Shapley algorithm produces a marriage that greatly favors the men at the expense of the women, or vice versa. The problem arises of finding a stable matching that is optimal under some more equitable or egalitarian criterion of optimality. This problem was posed by Knuth [6] and has remained unsolved for some time. Here, the objective of maximizing the average (or, equivalently, the total) “satisfaction” of all people is used. This objective is achieved when a person's satisfaction is measured by the position of his/her partner in his/her preference list. By exploiting the structure of the set of all stable matchings, and using graph-theoretic methods, an O ( n 4 ) algorithm for this problem is derived.

Stable Marriage and Roommates Problems with Restricted Edges: Complexity and Approximability

“…A stable matching satisfying all constraints on restricted edges exists if and only if there is a stable matching of weight −|Q| in the weighted instance, where Q is the set of forced edges. With the help of rotations, minimum weight stable matchings can be found in polynomial time [11][12][13][14] (see the final paragraph of Section 2 for more details on the role played by each of these references).…”

“…Irving et al [11] were the first to show that weighted sm can be solved in polynomial time, giving an O(n 4 log n) algorithm if the weight function is monotone in the preference ordering, non-negative and integral. Feder [12,14] showed a method to drop the monotonicity requirement.…”

Modeling Stable Matching Problems with Answer Set Programming

“…each set in a preference list has size at most 2). If there are no ties, the problem of finding an egalitarian or minimum regret stable set is solvable in polynomial time [13,10]. Since all stable sets consist of n couples in the classical SMP, the G-S algorithm trivially finds a maximum (or minimum) cardinality [7].…”

“…[10,13,20]). In this paper, we propose to use Answer Set Programming (ASP) as a general vehicle for modeling a large class of extensions and variations of the SMP.…”

“…The key rule which makes sure that I is a minimal model of the reduct is (15). The rules (13) imply that for each model of red(P crit , I) that does not contain sat, the literals of P ′crit ext in that model will correspond to a stable set of the SMP instance. In that case rule (15) will have a true body, since S is optimal, implying that sat should have been in the model.…”