Fooling Pairs in Randomized Communication Complexity

Shay Moran, Makrand Sinha, Amir Yehudayoff

Abstract: The fooling pairs method is one of the standard methods for proving lower bounds for deterministic two-player communication complexity. We study fooling pairs in the context of randomized communication complexity. We show that every fooling pair induces far away distributions on transcripts of private-coin protocols. We use the above to conclude that the private-coin randomized $$\varepsilon $$ -error communication complexity of a function f with a fooling set $$\mathcal S$$ is at least order $$\log \frac{\log |\mathcal S|}{\varepsilon }$$ . This relationship was earlier known to hold only for constant values of $$\varepsilon $$ . The bound we prove is tight, for example, for the equality and greater-than functions. As an application, we exhibit the following dichotomy: for every boolean function f and integer n, the (1/3)-error public-coin randomized communication complexity of the function $$\bigvee _{i=1}^{n}f(x_i,y_i)$$ is either at most c or at least n/c, where $$c>0$$ is a universal constant.