# 数学代写|图论代写graph theory代考|Stable Matching

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## 数学代写|图论代写graph theory代考|Stable Matching

In our discussion of matchings so far, we have only been concerned with finding the largest matching possible; but in many circumstances, one pairing may be preferable over another. When taking preferences into account, we no longer focus on whether two items can be paired but rather which pairing is best for the system. Initially, we will only consider those situations that can be modeled by a bipartite graph with an equal number of items in $X$ and $Y$. In addition, previous models demonstrated undesirable pairings by leaving off the edge in the graph; but with preferences added, we include every edge possible in the bipartite graph. This means the underlying graph will be a complete bipartite graph.

The preference model for matchings is often referred to as the Stable Marriage Problem to parallel Hall’s Marriage Theorem, and our terminology will reflect the marriage model. We start with two distinct, yet equal sized, groups of people, usually men and women, who have ranked the members of the other group. The stability of a matching is based on if switching two matched edges would result in happier couples.

## 数学代写|图论代写graph theory代考|Unacceptable Partners

Look back at the preferences in Example 5.13. By having each person rank all others of the opposite sex, we assume that all of these potential matches are acceptable. This is very clearly not accurate to a real world scenariosome people should never be married if even they are the only pair left. To adjust for this, we introduce the notion of an unacceptable partner. Consider the rankings below, where if a person is missing from the ranking, then they are deemed unacceptable (so Will is unacceptable to Diana and only Anne and Brenda are acceptable to Stan).

We are still looking for a matching in a bipartite graph, only now the graph is not complete. We must adjust our notion of a stable matching, since it is possible that not all people could be matched (think of a confirmed bachelor; he would label all women as unacceptable). Under these new conditions, a matching (with unacceptable partners) is stable so long as no unmatched pair $x$ and $y$ exist such that $x$ and $y$ are both acceptable to each other, and each is either single or prefers the other to their current partner. To account for this new definition of stable, we must make two minor adjustments to the Gale-Shapley Algorithm.

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