Inner Product and Set Disjointness

A major goal in complexity theory is to understand the communication complexity of number-on-the-forehead problems f :({0, 1} n ) k → {0, 1} with k > log n parties. We study the problems of inner product and set disjointness and determine their randomized communication complexity for every k ≥ log n , showing in both cases that Θ(1 + ⌈log n ⌉/ log ⌈1 + k / log n ⌉) bits are necessary and sufficient. In particular, these problems admit constant-cost protocols if and only if the number of parties is k ≥ n ϵ for some constant ϵ > 0.