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Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture

Authors Arkadev Chattopadhyay, Ankit Garg, Suhail Sherif

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Author Details

Arkadev Chattopadhyay
  • Tata Institute of Fundamental Research, Mumbai, India
Ankit Garg
  • Microsoft Research India, Bengaluru, India
Suhail Sherif
  • Vector Institute, Toronto, Canada

Funding

  • Sherif, Suhail: This work was mainly done while the author was at Microsoft Research India, Bengaluru.

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Arkadev Chattopadhyay, Ankit Garg, and Suhail Sherif. Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.FSTTCS.2021.13

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(n³) and randomized parity decision tree complexity Θ(n). This improves upon the recent work of Chattopadhyay, Mande and Sherif [Chattopadhyay et al., 2020] both qualitatively (in terms of designing a large number of examples) and quantitatively (shrinking 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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Communication complexity
Keywords
  • Approximate Rank
  • Randomized Parity Decision Trees
  • Randomized Communication Complexity
  • XOR functions
  • Subspace Designs

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