A Stable Marriage Requires Communication

The Gale-Shapely 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. Using a reduction to the communication complexity of the disjointness problem, we prove a significantly more general — albeit slightly weaker — result, showing that Ω(n2) Boolean queries of any type are required. Our lower bound generalizes to (A) randomized algorithms, (B) even just verifying the stability of a proposed marriage, (C) even allowing arbitrary separate preprocessing of the women’s preferences and of the men’s preferences, and (D) several variants of the basic problem, such as whether a given pair is married in every/some stable marriage.