Separating the k-party communication complexity hierarchy: an application of the Zarankiewicz problem

For every positive integer k, we construct an explicit family of functions f : {0, 1}n → {0, 1} which has (k+ 1)-party communication complexity O(k) under every partition of the input bits into k + 1 parts of equal size, and k-party communication complexity Ω ( n k42k ) under every partition of the input bits into k parts. This improves an earlier hierarchy theorem due to V. Grolmusz. Our construction relies on known explicit constructions for a famous open problem of Zarankiewicz: find the maximum number of edges in a graph on n vertices which does not contain Ks,t as a subgraph. The “Number on the Forehead” model of multiparty communication complexity was introduced by Chandra, Furst, and Lipton [2]. In this model, k players, P1, . . . , Pk, use a communication protocol to collaboratively evaluate a function f(x1, . . . , xk) on a particular input. The components xi of the input are written on the forehead of player Pi, so that each player sees the ∗Email: [email protected].