Probabilistic communication complexity over the reals

TL;DR

This paper explores communication complexity over the reals using polynomial exchanges, establishing bounds for recognizing geometric regions and problems like EMPTINESS and KNAPSACK.

Contribution

It introduces probabilistic communication protocols over the reals and provides sharp bounds for recognizing certain geometric regions and problems.

Findings

Sharp lower bound of 2n for recognizing the 2n-dimensional orthant.

Probabilistic complexity of recognizing the orthant does not exceed 4.

Lower bound of n/2 for recognizing certain polyhedra and unions of hyperplanes.

Abstract

Deterministic and probabilistic communication protocols are introduced in which parties can exchange the values of polynomials (rather than bits in the usual setting). It is established a sharp lower bound $2n$ on the communication complexity of recognizing the $2n$-dimensional orthant, on the other hand the probabilistic communication complexity of its recognizing does not exceed 4. A polyhedron and a union of hyperplanes are constructed in $\RR^{2n}$ for which a lower bound $n/2$ on the probabilistic communication complexity of recognizing each is proved. As a consequence this bound holds also for the EMPTINESS and the KNAPSACK problems.