A stable marriage requires communication

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 (n log n) 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 elaborate 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. Notably, unlike Segal's lower bound, our lower bound generalizes also to (A) randomized algorithms, (B) finding approximately-stable marriages (C) verifying the stability (or the approx-imate stability) of a proposed marriage, (D) allowing arbitrary separate preprocessing of the women's preferences profile and of the men's preferences profile, and (E) several variants of the basic problem, such as whether a given pair is married in every/some stable marriage.