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On the AC^0[oplus] Complexity of Andreev’s Problem

Author Aditya Potukuchi

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  • Theory of computation → Circuit complexity
Keywords
  • List Recovery
  • Sharp Threshold
  • Fourier Analysis

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Abstract

Andreev’s Problem is the following: Given an integer d and a subset of S subset F_q x F_q, is there a polynomial y = p(x) of degree at most d such that for every a in F_q, (a,p(a)) in S? We show an AC^0[oplus] lower bound for this problem. 
This problem appears to be similar to the list recovery problem for degree-d Reed-Solomon codes over F_q which states the following: Given subsets A_1,...,A_q of F_q, output all (if any) the Reed-Solomon codewords contained in A_1 x *s x A_q. In particular, we study this problem when the lists A_1, ..., A_q are randomly chosen, and are of a certain size. This may be of independent interest.

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Aditya Potukuchi. On the AC^0[oplus] Complexity of Andreev’s Problem. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.FSTTCS.2019.25

Author Details

Aditya Potukuchi
  • Department of Computer Science, Rutgers University, USA

Funding

  • Potukuchi, Aditya: Research supported in part by NSF grant CCF-1514164.

Acknowledgements

I would like to thank Swastik Kopparty for the discussions that led to Section 5, and Bhargav Narayanan for the discussions that led to Theorem 8.

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