On the complexity of trial and error
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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An instance of the stable marriage problem of size n involves n men and n women. Each participant ranks all members of the opposite sex in order of preference. A stable marriage is a complete matching M=((m/sub 1/, w/sub i1/), (m/sub 2/, w/sub i2/), . . ., (m/sub n/, w/sub in/)) such that no unmatched man and woman prefer each other to their partners in M. A pair (m/sub i/, w/sub j/) is stable if it is contained in some stable marriage. The problem of determining whether an arbitrary pair is stable in a given problem instance is studied. It is shown that the problem has a lower bound of Omega (n/sup 2/) in the worst case. As corollaries of the results, the lower bound of Omega (n/sup 2/) is established for several related stable marriage problems, including that of finding a stable marriage for any given problem instance.