Approximation ratio near 1 for SK model

Determine whether there exists a polynomial-time algorithm that, for the Sherrington–Kirkpatrick (SK) model with A drawn from the Gaussian Orthogonal Ensemble and objective maximize (1/(2n))⟨σ, Aσ⟩ over σ ∈ {+1,−1}^n, achieves an approximation ratio arbitrarily close to 1, i.e., for any fixed ε > 0 independent of n returns σ^alg such that (1/(2n))⟨σ^alg, Aσ^alg⟩ ≥ (1−ε)·OPT_n(A).

Background

In worst-case settings, binary quadratic optimization over the hypercube is hard to approximate beyond polylogarithmic factors, but for random SK instances semidefinite programming (SDP) attains an approximation ratio of about 0.834 relative to the Parisi value. This raises an algorithmic question specific to random SK instances: can one push the approximation ratio to 1−ε for any fixed ε?

The paper highlights this gap after comparing SDP’s performance with the Parisi prediction for the SK ground-state energy, noting that current convex relaxations—despite improvements in worst-case theory—do not close this gap on random SK instances.