Communication complexity of set-disjointness for all probabilities
We study set-disjointness in a generalized model of randomized two-party communication where the probability of acceptance must be at least α(n) on yes-inputs and at most β(n) on noinputs, for some functions α(n) > β(n). Our main result is a complete characterization of the private-coin communication complexity of set-disjointness for all functions α and β, and a nearcomplete characterization for public-coin protocols. In particular, we obtain a simple proof of a theorem of Braverman and Moitra (STOC 2013), who studied the case where α = 1/2 + ∈(n) and β = 1/2-∈(n). The following contributions play a crucial role in our characterization and are interesting in their own right. 1. We introduce two communication analogues of the classical complexity class that captures small bounded-error computations: we define a "restricted" class SBP (which lies between MA and AM) and an "unrestricted" class USBP. The distinction between them is analogous to the distinction between the well-known communication classes PP and UPP. 2. We show that the SBP communication complexity is precisely captured by the classical corruption lower bound method. This sharpens a theorem of Klauck (CCC 2003). 3. We use information complexity arguments to prove a linear lower bound on the USBP complexity of set-disjointness.
Leibniz International Proceedings in Informatics, LIPIcs
Göös, M., & Watson, T. (2014). Communication complexity of set-disjointness for all probabilities. Leibniz International Proceedings in Informatics, LIPIcs, 28, 721-736. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.721