Lower bounds for the stable marriage problem and its variants

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={(m₁, w_i1), (m₂, w_i2 ), . . ., (m_n, w_in)} such that no unmatched man and woman prefer each other to their partners in M. A pair (m_i, w_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 Ω(n²) in the worst case. As corollaries of the results, the lower bound of Ω(n²) is established for several related stable marriage problems, including that of finding a stable marriage for any given problem instance