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RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.65
Range Avoidance for Constant Depth Circuits: Hardness and Algorithms

Authors: Karthik Gajulapalli, Alexander Golovnev, Satyajeet Nagargoje, and Sidhant Saraogi

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract
Range Avoidance (Avoid) is a total search problem where, given a Boolean circuit 𝖢: {0,1}ⁿ → {0,1}^m, m > n, the task is to find a y ∈ {0,1}^m outside the range of 𝖢. For an integer k ≥ 2, NC⁰_k-Avoid is a special case of Avoid where each output bit of 𝖢 depends on at most k input bits. While there is a very natural randomized algorithm for Avoid, a deterministic algorithm for the problem would have many interesting consequences. Ren, Santhanam, and Wang (FOCS 2022) and Guruswami, Lyu, and Wang (RANDOM 2022) proved that explicit constructions of functions of high formula complexity, rigid matrices, and optimal linear codes, reduce to NC⁰₄-Avoid, thus establishing conditional hardness of the NC⁰₄-Avoid problem. On the other hand, NC⁰₂-Avoid admits polynomial-time algorithms, leaving the question about the complexity of NC⁰₃-Avoid open. We give the first reduction of an explicit construction question to NC⁰₃-Avoid. Specifically, we prove that a polynomial-time algorithm (with an NP oracle) for NC⁰₃-Avoid for the case of m = n+n^{2/3} would imply an explicit construction of a rigid matrix, and, thus, a super-linear lower bound on the size of log-depth circuits. We also give deterministic polynomial-time algorithms for all NC⁰_k-Avoid problems for m ≥ n^{k-1}/log(n). Prior work required an NP oracle, and required larger stretch, m ≥ n^{k-1}.
Cite as

Karthik Gajulapalli, Alexander Golovnev, Satyajeet Nagargoje, and Sidhant Saraogi. Range Avoidance for Constant Depth Circuits: Hardness and Algorithms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 65:1-65:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gajulapalli_et_al:LIPIcs.APPROX/RANDOM.2023.65,
  author =	{Gajulapalli, Karthik and Golovnev, Alexander and Nagargoje, Satyajeet and Saraogi, Sidhant},
  title =	{{Range Avoidance for Constant Depth Circuits: Hardness and Algorithms}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{65:1--65:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.65},
  URN =		{urn:nbn:de:0030-drops-188901},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.65},
  annote =	{Keywords: Boolean function analysis, Explicit Constructions, Low-depth Circuits, Range Avoidance, Matrix Rigidity, Circuit Lower Bounds}
}
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