Rigorous algorithmic and complexity-theoretic guarantees for the structured M‑layer lift

Establish precise algorithmic or complexity‑theoretic performance guarantees for the structured M‑layer lift, for example proving polynomial‑time convergence to global minima (maximum‑a‑posteriori configurations) or other formal bounds on optimization performance across spin‑glass Ising models and related combinatorial problems.

Background

The paper reports empirical improvements in reaching near‑MAP configurations using the structured M‑layer lift across diverse benchmarks, but explicitly refrains from claiming polynomial‑time guarantees for global optimization. The authors highlight that sharper theoretical statements—such as formal runtime bounds, convergence guarantees, or complexity‑theoretic characterizations—are not yet established.

This open question asks for rigorous results that quantify the algorithmic power and limitations of the structured lift, moving beyond empirical evidence to formal proofs or bounds.

References

While spin-level and cavity-level simulations reach near-MAP configurations on all tested instances, we do not claim polynomial-time guarantees for finding global minima in general. The simplicity of the construction suggests that sharper algorithmic or complexity-theoretic statements may be achievable, but such results remain open.

— Reshaping Global Loop Structure to Accelerate Local Optimization by Smoothing Rugged Landscapes  (2602.01490 - Leleu et al., 1 Feb 2026) in Section VI (Conclusion)