Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture

TL;DR

This paper constructs numerous Boolean functions with low approximate Fourier sparsity but high randomized parity decision tree complexity, providing stronger counterexamples to the log-approximate-rank conjecture and highlighting gaps between decision tree and communication complexities.

Contribution

It introduces a large class of functions with improved separations, advancing the understanding of complexity measures related to the log-approximate-rank conjecture.

Findings

Constructed functions with approximate Fourier sparsity at most O(n^3)

Achieved randomized parity decision tree complexity of Θ(n)

Improved the gap from quartic to cubic compared to prior work

Abstract

We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on $n$ input bits, each of which has approximate Fourier sparsity at most $O(n^3)$ and randomized parity decision tree complexity $\Theta(n)$. This improves upon the recent work of Chattopadhyay, Mande and Sherif (JACM '20) both qualitatively (in terms of designing a large number of examples) and quantitatively (improving the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).