A stable marriage requires communication

In 1976, Knuth asked whether the worst-case running-time of the Gale-Shapley algorithm for the Stable Marriage Problem can be improved when non-sequential access to the input is allowed. Partial negative answers were given by Ng and Hirschberg and as part of Segal's general communication-complexity analysis. We give a far simpler, yet significantly more powerful, argument showing that Ω(n²) Boolean queries of any type are required for finding a stable — or even approximately stable — marriage. Unlike Segal's lower bound, our lower bound generalizes additionally to (A) randomized algorithms, (B) allowing arbitrary separate preprocessing of the women's and men's respective preferences profiles, (C) related problems, e.g. whether a given pair is married in every/some stable marriage, (D) whether a proposed marriage is stable or far from stable. To analyze “approximately stable” marriages, we introduce the notion of “distance to stability” and provide an efficient algorithm for its computation.