This paper deals with a strategic issue in the stable marriage model with complete preference lists (i.e., a preference list of an agent is a permutation of all the members of the opposite sex). Given complete preference lists of n men over n women, and a marriage µ, we consider the problem for finding preference lists of n women over n men such that the men-proposing deferred acceptance algorithm (Gale-Shapley algorithm) adopted to the lists produces µ. We show a simple necessary and sufficient condition for the existence of a set of preference lists of women over men. Our condition directly gives an O(n2) time algorithm for finding a set of preference lists, if it exists.

An instance of the classical stable marriage problem requires all participants to submit a strictly ordered preference list containing all members of the opposite sex. However, considering applications in real-world, we can think of two natural relaxations, namely, incomplete preference lists and ties in the lists. Either variation leaves the problem polynomially solvable, but it is known that finding a maximum cardinality stable matching is NP-hard when both variations are allowed. It is easy to see that 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. A few approximation algorithms have been proposed with approximation ratio better than two, but they are only for restricted instances, such as restricting occurrence of ties and/or lengths of ties. Up to the present, there is no known approximation algorithm with ratio better than two for general inputs. In this paper, we give the first nontrivial result for approximation of factor less than two for general instances. Our algorithm achieves the ratio for an arbitrarily positive constant c, where N denotes the number of men in an input.

The stable marriage problem is one of the basic problems proposed in 1962. In this paper, we consider a distributed stable marriage problem. This problem is applicable to cooperative works of autonomous robots in distributed environments. We show a Gale-Shapley based protocol to obtain stable matching and introduce autonomous mechanism for exchanging partners, called divorce process, in distributed environments. We report some interesting results of matching games by computer simulation.

In this paper, we propose the distributed stable marriage problem and apply it to planning for cooperative works of autonomous mobile robots and battery charger stations. We develop and analyze a distributed algorithm to determine the partner by message communication.