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.ICALP.2023.28
URN: urn:nbn:de:0030-drops-180801
URL: https://drops.dagstuhl.de/opus/volltexte/2023/18080/
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Bogdanov, Andrej ; Rosen, Alon

Nondeterministic Interactive Refutations for Nearest Boolean Vector

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LIPIcs-ICALP-2023-28.pdf (0.7 MB)


Abstract

Most n-dimensional subspaces 𝒜 of ℝ^m are Ω(√m)-far from the Boolean cube {-1, 1}^m when n < cm for some constant c > 0. How hard is it to certify that the Nearest Boolean Vector (NBV) is at least γ √m far from a given random 𝒜?
Certifying NBV instances is relevant to the computational complexity of approximating the Sherrington-Kirkpatrick Hamiltonian, i.e. maximizing x^TAx over the Boolean cube for a matrix A sampled from the Gaussian Orthogonal Ensemble. The connection was discovered by Mohanty, Raghavendra, and Xu (STOC 2020). Improving on their work, Ghosh, Jeronimo, Jones, Potechin, and Rajendran (FOCS 2020) showed that certification is not possible in the sum-of-squares framework when m ≪ n^1.5, even with distance γ = 0.
We present a non-deterministic interactive certification algorithm for NBV when m ≫ n log n and γ ≪ 1/mn^1.5. The algorithm is obtained by adapting a public-key encryption scheme of Ajtai and Dwork.

BibTeX - Entry

@InProceedings{bogdanov_et_al:LIPIcs.ICALP.2023.28,
  author =	{Bogdanov, Andrej and Rosen, Alon},
  title =	{{Nondeterministic Interactive Refutations for Nearest Boolean Vector}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{28:1--28:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18080},
  URN =		{urn:nbn:de:0030-drops-180801},
  doi =		{10.4230/LIPIcs.ICALP.2023.28},
  annote =	{Keywords: average-case complexity, statistical zero-knowledge, nondeterministic refutation, Sherrington-Kirkpatrick Hamiltonian, complexity of statistical inference, lattice smoothing}
}

Keywords: average-case complexity, statistical zero-knowledge, nondeterministic refutation, Sherrington-Kirkpatrick Hamiltonian, complexity of statistical inference, lattice smoothing
Collection: 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)
Issue Date: 2023
Date of publication: 05.07.2023


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