Stable Roommates Research Articles

Stable matchings and stable partitions∗

Recently Tan [7] defined a new structure for the stable roommates problem, called a “stable partition” which is a generalization of the notion of the stable matching. He proved that every instance of that problem contains at least one such structure, and obtained a succinct certificate of the non-existence of a stable matching. In this paper. we establish several results on properties of the stable partitions, present a simple proof of a recent theorem of Tan, and solve a maximum stable matching problem.

A maximum stable matching for the roommates problem

The stable roommates problem is that of matchingn people inton/2 disjoint pairs so that no two persons, who are not paired together, both prefer each other to their respective mates under the matching. Such a matching is called “a complete stable matching”. It is known that a complete stable matching may not exist. Irving proposed anO(n 2) algorithm that would find one complete stable matching if there is one, or would report that none exists. Since there may not exist any complete stable matching, it is natural to consider the problem of finding a maximum stable matching, i.e., a maximum number of disjoint pairs of persons such that these pairs are stable among themselves. In this paper, we present anO(n 2) algorithm, which is a modified version of Irving's algorithm, that finds a maximum stable matching.

The Structure of the Stable Roommate Problem: Efficient Representation and Enumeration of All Stable Assignments

The stable roommates problem is a well-known problem of matching $2n$ people into disjoint pairs to achieve a certain type of stability. The problem strictly generalizes the better-known stable marriage problem. It has been previously shown [Irving and Leather, SIAM J. Comput., 15 (1986), pp. 655–667], that the set of stable marriages, for a given instance, can be compactly represented and this representation has been exploited to yield a number of very efficient algorithms concerned with stable marriage [Irving, Leather, and Gusfield, J. Assoc. Comput. Mach., 34 (1987), pp. 532–543], [Gusfield, SIAM J. Comput., 16 (1987), pp. 111–128], [Gusfield et al., J. Combin. Theory, Ser A., (1987), pp. 304–309]. In this paper, we generalize the structure of the stable marriages to obtain two efficiently computed, small, implicit representations of the set of all stable roommate assignments, for any given instance. One representation is a partial order $\Pi $ on $O(n^2 )$ elements such that the stable assignments ar...

Stable matching with preferences derived from a psychological model

We study a special case of the Stable Roommates problem in which preferences are derived from a psychological model common in social choice literature. When preferences are ‘single-peaked’ and ‘narcissistic’, there exists a unique stable matching, and it can be constructed in O( n) time. We also show how to recognize quickly, when a set of preferences is single-peaked.

An efficient algorithm for the “stable roommates” problem

The stable marriage problem is that of matching n men and n women, each of whom has ranked the members of the opposite sex in order of preference, so that no unmatched couple both prefer each other to their partners under the matching. At least one stable matching exists for every stable marriage instance, and efficient algorithms for finding such a matching are well known. The stable roommates problem involves a single set of even cardinality n, each member of which ranks all the others in order of preference. A stable matching is now a partition of this single set into n 2 pairs so that no two unmatched members both prefer each other to their partners under the matching. In this case, there are problem instances for which no stable matching exists. However, the present paper describes an O( n 2) algorithm that will determine, for any instance of the problem, whether a stable matching exists, and if so, will find such a matching.

Total Ordering Problem

The problem of finding a total ordering of a finite set satisfying a given set of in-between restrictions is considered. It is shown that the problem is $NP$-complete.