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Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture

Authors Arkadev Chattopadhyay, Ankit Garg, Suhail Sherif

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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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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).

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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

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.

References

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