Lower Bounds for Number-in-Hand Multiparty Communication Complexity, Made Easy

TL;DR

This paper introduces a new symmetrization technique to establish tight lower bounds on randomized multiparty communication complexity in both blackboard and message-passing models, applicable to various problems including XOR, AND, and graph connectivity.

Contribution

The paper presents a natural, intuitive symmetrization method for proving lower bounds, simplifying analysis and extending results to multiple problems in multiparty communication complexity.

Findings

Proved a tight nk lower bound for XOR in the blackboard model.

Established nk lower bound for AND in message-passing model under certain conditions.

Demonstrated applicability to other problems like majority and graph connectivity.

Abstract

In this paper we prove lower bounds on randomized multiparty communication complexity, both in the \emph{blackboard model} (where each message is written on a blackboard for all players to see) and (mainly) in the \emph{message-passing model}, where messages are sent player-to-player. We introduce a new technique for proving such bounds, called \emph{symmetrization}, which is natural, intuitive, and often easy to use. For example, for the problem where each of $k$ players gets a bit-vector of length $n$, and the goal is to compute the coordinate-wise XOR of these vectors, we prove a tight lower bounds of $\Omega(nk)$ in the blackboard model. For the same problem with AND instead of XOR, we prove a lower bounds of roughly $\Omega(nk)$ in the message-passing model (assuming $k \le n/3200$) and $\Omega(n \log k)$ in the blackboard model. We also prove lower bounds for bit-wise majority, for a graph-connectivity problem, and for other problems; the technique seems applicable to a wide range of other problems as well. All of our lower bounds allow randomized communication protocols with two-sided error. We also use the symmetrization technique to prove several direct-sum-like results for multiparty communication.