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/
Go to the corresponding LIPIcs Volume Portal

Guruswami, Venkatesan ; Lyu, Xin ; Wang, Xiuhan

Range Avoidance for Low-Depth Circuits and Connections to Pseudorandomness

pdf-format:
LIPIcs-APPROX20.pdf (0.8 MB)

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: Pseudorandomness, Explicit constructions, Low-depth circuits, Boolean function analysis, Hitting sets
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)
Issue Date: 2022
Date of publication: 15.09.2022

DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI