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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.
The algorithm can be implemented to run in time quadratic in the number of participants, and linear in the size of the input to the algorithm. The stable matching problem, and the Gale–Shapley algorithm solving it, have widespread real-world applications, including matching American medical students to residencies and French university ...
In any stable table, if every reduced list contains exactly one individual, then pairing each individual with the single person on their list gives a stable matching. If the stable roommates problem instance has a stable matching, then there is a stable matching contained in any one of the stable tables. Any stable subtable of a stable table ...
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 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 .
The same is true of the family of all fractional stable matchings, the points of the stable matching polytope. [3] In the stable matching polytope, one can define one matching to dominate another if, for every doctor and hospital, the total fractional value assigned to matches for that doctor that are at least as good (for the doctor) as that ...
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):
They adapted the David Gale's top-trading-cycle algorithm to allow the one-sided matching with waiting-list options, and proposed efficient and incentive-compatible chain selection rules. The same team subsequently showed that efficient outcomes with good incentive properties can be found in computationally efficient ways when only pairwise ...