Evaluating the ground state energy _n(A; Σ_n) for detection

Develop computationally tractable methods to evaluate or tightly approximate the ground state energy _n(A_G; Σ_n)=max_{σ∈Σ_n}(1/(2n))⟨σ, A_Gσ⟩ needed by the Parisi-based detection procedure for community structure, with guarantees applicable to relevant random graph models.

Background

The detection method requires computing the maximum of the binary quadratic form associated with minimum bisection. The paper explicitly notes that, in general, this quantity is not computationally available, motivating algorithmic work later in the paper to approximate it.

Closing this gap is essential for making the Parisi-based detection test operational in high-dimensional statistical settings such as the stochastic block model.

References

Of course, this approach runs into two difficulties: $(i)$~We do not know how to set $\delta_n$ (from a statistics perspective, Theorem \ref{thm:parisi} merely says that any positive constant will work for $n$ large enough); $(ii)$~In general, we do not know how to evaluate $\mathsf{OPT}_n(A;\Sigma_n)$. Some of the developments discussed in Section \ref{sec:Algo} address the last problem.

— Spin Glass Concepts in Computer Science, Statistics, and Learning  (2602.23326 - Montanari, 26 Feb 2026) in Section 2 (Parisi’s formula), final remark on detection using Parisi’s value