Communication Lower Bounds Using Dual Polynomials

TL;DR

This paper surveys recent advances in communication complexity lower bounds using dual polynomials, highlighting key theorems, methods, and extensions to multiparty models with implications for complexity class separations.

Contribution

It provides a unified overview of dual polynomial techniques and their applications in deriving communication complexity lower bounds, including new multiparty results.

Findings

Sherstov's Degree/Discrepancy Theorem links threshold degree to discrepancy.

Two methods for lower bounds based on approximate degree are detailed.

Extended pattern matrix method yields improved multiparty lower bounds for DISJOINTNESS.

Abstract

Representations of Boolean functions by real polynomials play an important role in complexity theory. Typically, one is interested in the least degree of a polynomial p(x_1,...,x_n) that approximates or sign-represents a given Boolean function f(x_1,...,x_n). This article surveys a new and growing body of work in communication complexity that centers around the dual objects, i.e., polynomials that certify the difficulty of approximating or sign-representing a given function. We provide a unified guide to the following results, complete with all the key proofs: (1) Sherstov's Degree/Discrepancy Theorem, which translates lower bounds on the threshold degree of a Boolean function into upper bounds on the discrepancy of a related function; (2) Two different methods for proving lower bounds on bounded-error communication based on the approximate degree: Sherstov's pattern matrix method and Shi and Zhu's block composition method; (3) Extension of the pattern matrix method to the multiparty model, obtained by Lee and Shraibman and by Chattopadhyay and Ada, and the resulting improved lower bounds for DISJOINTNESS; (4) David and Pitassi's separation of NP and BPP in multiparty communication complexity for k=(1-eps)log n players.