Stable Marriage Problem: a Classic Matching Problem and Algorithm

Stable marriage problem is a problem of classic computer science. In can be found as a problem in mathematics as well. This classical matching problem is first introduced by Gale and Shapely. This problem has the purpose to find a stable matching, especially between two sets of elements that are given a collection of preferences for each elements. This problem is considered as classic because it can be found in the system of computer, and computer is a technology that everybody has been knowing for decades. There assists a polyominal time algorithm to find the solution  of this matching problem.

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If we are studying this matchmaking problem further, we will find this deferred acceptance of stable marriage problem algorithm.

We might reckon a set of women and men, where each of them has a list of some members of their opposite sex as their preference list. The list is ordered. In every process, what is possible to happen is a single man proposes to the the first woman who is listed on his tilt. Yet, the woman has not been proposed by him before. The second case, a woman chooses to reject her current engagement with a man because she prefers to accept the new proposal that she receive from another man. This kind of problem is also known as the men-proposing algorithm.

There are three theorems that can be considered as the completion of that problem of matchmaking, and to make the marriage become stable at the same time. The first theory is the order of proposal of engagement will not affect the stable matching which is made by the men-proposing algorithm. The second theorem is that the matching made by the mentioned men-proposing algorithm is considered as the most stable matching for men. However, on the contrary, it is he most unstable matching for women. This second theorem is called the men-optimal matching. Lastly, the third classical result is in every stable matching, the number of people who remain unengaged is stable or the same.

The structure of proof of this matchmaking problem is divided into three steps. The first step is am algorithm counting the number of the stable man of a given woman. The next step is to bound the possibility of having more than a single stable husband. The last step for this stable marriage problem is to bound the unengaged people by the solution of the occupancy problematic.