In this note, we demonstrate that the problem of “many-to-one matching with (strict) preferences over colleagues ” is actually more difficult than the classical many-to-one matching problem, “matching without preferences over colleagues. ” We give an explicit reduction of any problem of the latter type to a problem of the former type. This construction leads to the first algorithm which finds all stable matchings in the setting of “matching without preferences over colleagues, ” for any set of preferences. Our construction directly extends to generalized matching settings
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