Communication Complexity is NP-hard

TL;DR

This paper proves that determining whether the communication complexity of a function is below a certain threshold is an NP-hard problem, establishing its computational difficulty.

Contribution

It demonstrates that deciding if a function's communication complexity is at most a given value is NP-hard, resolving a long-standing open question.

Findings

Deciding CC(f) ≤ k is NP-hard.

The problem is in NP but also NP-hard.

Establishes computational difficulty of communication complexity decision.

Abstract

In the paper where he first defined Communication Complexity, Yao asks: \emph{Is computing $CC(f)$ (the 2-way communication complexity of a given function $f$) NP-complete?} The problem of deciding whether $CC(f) \le k$, when given the communication matrix for $f$ and a number $k$, is easily seen to be in NP. Kushilevitz and Weinreb have shown that this problem is cryptographically hard. Here we show it is NP-hard.