eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-07-05
28:1
28:14
10.4230/LIPIcs.ICALP.2023.28
article
Nondeterministic Interactive Refutations for Nearest Boolean Vector
Bogdanov, Andrej
1
https://orcid.org/0000-0002-0338-6151
Rosen, Alon
2
3
https://orcid.org/0000-0002-3021-7150
University of Ottawa, Canada
Bocconi University, Milano, Italy
Reichman University, Herzliya, Israel
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol261-icalp2023/LIPIcs.ICALP.2023.28/LIPIcs.ICALP.2023.28.pdf
average-case complexity
statistical zero-knowledge
nondeterministic refutation
Sherrington-Kirkpatrick Hamiltonian
complexity of statistical inference
lattice smoothing