Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture

Arkadev Chattopadhyay; Ankit Garg; Suhail Sherif

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arXiv · 2009.02717

Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture

Arkadev Chattopadhyay·Ankit Garg·Suhail Sherif

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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).

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Arkadev Chattopadhyay, Ankit Garg, Suhail Sherif. 2020-09-06. Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture. https://arxiv.org/abs/2009.02717

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