Choosing, agreeing, and eliminating in communication complexity

Abstract

We consider several questions inspired by the direct-sum problem in (two-party) communication complexity. In all questions, there are k fixed Boolean functions f1,...,fk and Alice and Bob have k inputs x1,...,xk and y1,...,yk , respectively. In the eliminate problem, Alice and Bob should output a vector σ1,...,σk such that fi (x i ) ≠ σi for at least one i (i.e., their goal is to eliminate one of the 2k output vectors); in choose, Alice and Bob should return (i,fi (xi, yi )) and in agree they should return fi (xi, yi ), for some i. The question, in each of the three cases, is whether one can do better than solving one (say, the first) instance. We study these three problems and prove various positive and negative results. © 2010 Springer-Verlag Berlin Heidelberg.