A Note on the LogRank Conjecture in Communication Complexity

TL;DR

This paper introduces a new communication protocol related to the LogRank conjecture in communication complexity, using low-degree multilinear polynomial representations to approximate the function with a bound similar to the conjecture.

Contribution

The paper presents a protocol that computes a related quantity to the function using a bound similar to the conjecture, leveraging low-degree multilinear polynomial representations.

Findings

Protocol computes a related quantity to the original function

Uses low-degree multilinear polynomial representations

Provides insights that may help resolve the conjecture

Abstract

The LogRank conjecture of Lov\'asz and Saks from 1988 is the most famous open problem in the communication complexity theory. The statement is as follows: Suppose that two players intend to compute a Boolean function $f(x,y)$ when $x$ is known for the first and $y$ for the second player, and they may send and receive messages encoded with bits, then they can compute $f(x,y)$ with exchanging $(\log \rank (M_f))^c $ bits, where $M_f$ is a Boolean matrix, determined by function $f$. The problem is widely open and very popular, and it has resisted numerous attacks in the last 35 years. The best upper bound is still exponential in the bound of the conjecture. Unfortunately, we cannot prove the conjecture, but we present a communication protocol with $(\log \rank (M_f))^c $ bits, which computes a -- somewhat -- related quantity to $f(x,y)$. The relation is characterized by a representation of low-degree, multi-linear polynomials modulo composite numbers. This result of ours may help to settle this long-time open conjecture.