Suboptimality of stable algorithms for all p-spin Ising models with p≥3

Determine whether, for the p-spin Ising spin glass model with p≥3, the largest ground-state value achievable by stable (noise-insensitive) polynomial-time algorithms is strictly smaller than the Parisi-optimal value, equivalently whether the extended variational value η_{p,ALG}=inf_{μ∈ℒ}P(μ) remains strictly less than η_{p,OPT}=inf_{μ∈𝒰}P(μ) for all such p.

Background

The paper shows that when p≥4 is even, there is a strict inequality between the extended variational value over a larger class ℒ and the Parisi value over the original class 𝒰, implying a fundamental barrier for stable algorithms (via Branching-OGP) and identifying the tight algorithmic threshold η_{p,ALG}.

The authors conjecture that this strict suboptimality of stable algorithms persists for all p≥3 (irrespective of parity). Proving this would extend the current tight characterization of algorithmic tractability to all p≥3 and further solidify the role of Overlap Gap Property structures as barriers to efficient algorithms beyond the even p≥4 regime.

References

On the other hand, when $p\ge 4$ is even, the largest value achievable by stable algorithms is strictly sub-optimal. It is conjectured that this remains the case for all $p\ge 3$ regardless of the parity and has been verified for the case when $p$ is sufficiently large.

— Turing in the shadows of Nobel and Abel: an algorithmic story behind two recent prizes (2501.15312 - Gamarnik, 25 Jan 2025) in Section “Branching-OGP. Tight characterization of algorithmically solvable spin glasses”