Range Avoidance for Constant Depth Circuits: Hardness and Algorithms
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}.
Boolean function analysis
Explicit Constructions
Low-depth Circuits
Range Avoidance
Matrix Rigidity
Circuit Lower Bounds
Theory of computation~Circuit complexity
65:1-65:18
RANDOM
https://arxiv.org/abs/2303.05044
We would like to thank Justin Thaler, Sam King, and anonymous reviewers for their helpful comments on our paper.
Karthik
Gajulapalli
Karthik Gajulapalli
Georgetown University, Washington, DC, USA
https://kgajulapalli.org
Alexander
Golovnev
Alexander Golovnev
Georgetown University, Washington, DC, USA
https://golovnev.org
Satyajeet
Nagargoje
Satyajeet Nagargoje
Georgetown University, Washington, DC, USA
https://satyajeetn.github.io
Sidhant
Saraogi
Sidhant Saraogi
Georgetown University, Washington, DC, USA
https://sarsid.github.io
10.4230/LIPIcs.APPROX/RANDOM.2023.65
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Karthik Gajulapalli, Alexander Golovnev, Satyajeet Nagargoje, and Sidhant Saraogi
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