Communication Complexity Theory: Thirty-Five Years of Set Disjointness

The set disjointness problem features k communicating parties and k subsets S1, S2, . . . , Sk ⊆ {1, 2, . . . , n}. No single party knows all k subsets, and the objective is to determine with minimal communication whether the k subsets have nonempty intersection. The important special case k = 2 corresponds to two parties trying to determine whether their respective sets intersect. The study of the set disjointness problem spans almost four decades and offers a unique perspective on the remarkable evolution of communication complexity theory. We discuss known results on the communication complexity of set disjointness in the deterministic, nondeterministic, randomized, unbounded-error, and multiparty models, emphasizing the variety of mathematical techniques involved.