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With this condition, a stable matching will still exist, and can still be found by the Gale–Shapley algorithm. For this kind of stable matching problem, the rural hospitals theorem states that: The set of assigned doctors, and the number of filled positions in each hospital, are the same in all stable matchings.
A stable matching always exists, and the algorithmic problem solved by the Gale–Shapley algorithm is to find one. [3] The stable matching problem has also been called the stable marriage problem, using a metaphor of marriage between men and women, and many sources describe the Gale–Shapley algorithm in terms of marriage proposals. However ...
The algorithm will determine, for any instance of the problem, whether a stable matching exists, and if so, will find such a matching. Irving's algorithm has O( n 2 ) complexity , provided suitable data structures are used to implement the necessary manipulation of the preference lists and identification of rotations.
The lattice of stable matchings is based on the following weaker structure, a partially ordered set whose elements are the stable matchings. Define a comparison operation on the stable matchings, where if and only if all doctors prefer matching to matching : either they have the same assigned hospital in both matchings, or they are assigned a better hospital in than they are in .
It also has a unique smallest element, the integer stable matching found by a version of the Gale–Shapley algorithm in which the hospitals make the proposals. [ 3 ] Consistently with this partial order, one can define the meet of two fractional matchings to be a fractional matching that is as low as possible in the partial order while ...
The algorithm computes perfect matching between sets of men and women, thus finding the critical set of men who are engaged to multiple women. Since such engagements are never stable, all such pairs are deleted and the proposal sequence will be repeated again until either 1) some man's preference list becomes empty (in which case no strongly ...
The algorithm must terminate, since in each iteration we remove at least one agent. It can be proved that this algorithm leads to a core-stable allocation. For example, [ 2 ] : 223–224 suppose the agents' preference ordering is as follows (where only the at most 4 top choices are relevant):
A matching M of a graph G is maximal if every edge in G has a non-empty intersection with at least one edge in M. The following figure shows examples of maximal matchings (red) in three graphs. A maximum matching (also known as maximum-cardinality matching [2]) is a matching that contains the largest possible number of edges. There may be many ...