Assuming worst-case lattice hardness, no polynomial-time algorithm can solve SBP at kappa close to 1/sqrt(x) or NPP at kappa close to 2^{-log^3 m}.
Proof of the contiguity conjecture and lognormal limit for the symmetric perceptron, 2021
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Symmetric Perceptrons, Number Partitioning and Lattices
Assuming worst-case lattice hardness, no polynomial-time algorithm can solve SBP at kappa close to 1/sqrt(x) or NPP at kappa close to 2^{-log^3 m}.