License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.20
URN: urn:nbn:de:0030-drops-171428
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/17142/
Guruswami, Venkatesan ;
Lyu, Xin ;
Wang, Xiuhan
Range Avoidance for Low-Depth Circuits and Connections to Pseudorandomness
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.
BibTeX - Entry
@InProceedings{guruswami_et_al:LIPIcs.APPROX/RANDOM.2022.20,
author = {Guruswami, Venkatesan and Lyu, Xin and Wang, Xiuhan},
title = {{Range Avoidance for Low-Depth Circuits and Connections to Pseudorandomness}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
pages = {20:1--20:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-249-5},
ISSN = {1868-8969},
year = {2022},
volume = {245},
editor = {Chakrabarti, Amit and Swamy, Chaitanya},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/17142},
URN = {urn:nbn:de:0030-drops-171428},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.20},
annote = {Keywords: Pseudorandomness, Explicit constructions, Low-depth circuits, Boolean function analysis, Hitting sets}
}
Keywords: |
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Pseudorandomness, Explicit constructions, Low-depth circuits, Boolean function analysis, Hitting sets |
Collection: |
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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022) |
Issue Date: |
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2022 |
Date of publication: |
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15.09.2022 |