The stable roommates problem is similar to the stable marriage problem, but differs in that all participants belong to a single pool (instead of being divided into equal numbers of "men" and "women").
The hospitals/residents problem — also known as the college admissions problem — differs from the stable marriage problem in that the "women" can accept "proposals" from more than one "man" (e.g., a hospital can take multiple residents, or a college can take an incoming class of more than one student).
It runs in The assignment problem seeks to find a matching in a weighted bipartite graph that has maximum weight.
Maximum weighted matchings do not have to be stable, but in some applications a maximum weighted matching is better than a stable one.
A matching is a mapping from the elements of one set to the elements of the other set.
A matching is not stable if: In other words, a matching is stable when there does not exist any match (A, B) by which both A and B would be individually better off than they are with the element to which they are currently matched.
A matching will be called weakly stable unless there is a couple each of whom strictly prefers the other to his/her partner in the matching. Irving A matching is strongly stable if there is no couple x, y such that x strictly prefers y to his/her partner and y either strictly prefers x to his/her partner or is indifferent between them. Irving has provided the algorithm which checks if such strongly stable matching exists and outputs the matching if it exists.
The algorithm computes perfect matching between sets of men and women, thus finding the critical set of men who are engaged to multiple women.
If Alice accepted his proposal, yet is not married to him at the end, she must have dumped him for someone she likes more, and therefore doesn't like Bob more than her current partner.Let Alice and Bob both be engaged, but not to each other.Upon completion of the algorithm, it is not possible for both Alice and Bob to prefer each other over their current partners.Algorithms to solve the hospitals/residents problem can be hospital-oriented (female-optimal) or resident-oriented (male-optimal).This problem was solved, with an algorithm, in the same original paper by Gale and Shapley, in which the stable marriage problem was solved.