The Gale-Shapley algorithm for the Stable Marriage Problem is known to take
Θ(n2) steps to find a stable marriage in the worst case, but only
Θ(nlogn) steps in the average case (with n women and n men). In
1976, Knuth asked whether the worst-case running time can be improved in a
model of computation that does not require sequential access to the whole
input. A partial negative answer was given by Ng and Hirschberg, who showed
that Θ(n2) queries are required in a model that allows certain natural
random-access queries to the participants' preferences. A significantly more
general - albeit slightly weaker - lower bound follows from Segal's general
analysis of communication complexity, namely that Ω(n2) Boolean queries
are required in order to find a stable marriage, regardless of the set of
allowed Boolean queries.
Using a reduction to the communication complexity of the disjointness
problem, we give a far simpler, yet significantly more powerful argument
showing that Ω(n2) Boolean queries of any type are indeed required for
finding a stable - or even an approximately stable - marriage. Notably, unlike
Segal's lower bound, our lower bound generalizes also to (A) randomized
algorithms, (B) allowing arbitrary separate preprocessing of the women's
preferences profile and of the men's preferences profile, (C) several variants
of the basic problem, such as whether a given pair is married in every/some
stable marriage, and (D) determining whether a proposed marriage is stable or
far from stable. In order to analyze "approximately stable" marriages, we
introduce the notion of "distance to stability" and provide an efficient
algorithm for its computation