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Range Avoidance for Low-Depth Circuits and Connections to Pseudorandomness

Authors Venkatesan Guruswami, Xin Lyu, Xiuhan Wang

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Author Details

Venkatesan Guruswami
  • University of California Berkeley, CA, USA
Xin Lyu
  • University of California Berkeley, CA, USA
Xiuhan Wang
  • Tsinghua University, Beijing, China

Funding

  • Guruswami, Venkatesan: Research supported in part by NSF CCF-2210823 and a Simons Investigator Award.

Acknowledgements

We thank anonymous reviewers for helpful comments and suggestions.

Cite As Get BibTex

Venkatesan Guruswami, Xin Lyu, and Xiuhan Wang. Range Avoidance for Low-Depth Circuits and Connections to Pseudorandomness. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 20:1-20:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.20

Abstract

In the range avoidance problem, the input is a multi-output Boolean circuit with more outputs than inputs, and the goal is to find a string outside its range (which is guaranteed to exist). We show that well-known explicit construction questions such as finding binary linear codes achieving the Gilbert-Varshamov bound or list-decoding capacity, and constructing rigid matrices, reduce to the range avoidance problem of log-depth circuits, and by a further recent reduction [Ren, Santhanam, and Wang, FOCS 2022] to NC⁰₄ circuits where each output depends on at most 4 input bits. 
On the algorithmic side, we show that range avoidance for NC⁰₂ circuits can be solved in polynomial time. We identify a general condition relating to correlation with low-degree parities that implies that any almost pairwise independent set has some string that avoids the range of every circuit in the class. We apply this to NC⁰ circuits, and to small width CNF/DNF and general De Morgan formulae (via a connection to approximate-degree), yielding non-trivial small hitting sets for range avoidance in these cases.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Pseudorandomness
  • Explicit constructions
  • Low-depth circuits
  • Boolean function analysis
  • Hitting sets

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