Theory of Computing
-------------------
Title : Communication Complexity of Set-Disjointness for All Probabilities
Authors : Mika Goos and Thomas Watson
Volume : 12
Number : 9
Pages : 1-23
URL : https://theoryofcomputing.org/articles/v012a009
Abstract
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We study set-disjointness in a generalized model of randomized two-
party communication where the probability of acceptance must be at
least $\alpha(n)$ on yes-inputs and at most $\beta(n)$ on no-inputs,
for some functions $\alpha(n)> \beta(n)$. Our main result is a
complete characterization of the private-coin communication complexity
of set-disjointness for all functions $\alpha$ and $\beta$, and a
near-complete 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
$\alpha=1/2+\epsilon(n)$ and $\beta=1/2-\epsilon(n)$. 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.
A preliminary version of this paper appeared in the Proceedings of
the 18th International Workshop on Randomization and Computation
(RANDOM), 2014.