Critical small-set vertex expansion threshold in bipartite random regular graphs

Ascertain the computational hardness of certifying the exact critical bound \(\Phi_{\varepsilon}^{v}(G) \ge d/2\) (with no error terms) for bipartite random d-regular graphs \(G \sim \mathcal{G}((n/2,n/2), d)\) when \(d=q+1\) (q a prime power) in the limit \(\varepsilon \to 0\). Determine whether polynomial-time certification at this threshold is possible or provably hard.

Background

Using explicit bipartite Ramanujan constructions, the authors show (conditional on their conjecture) that Kahale’s spectral lower bound for small-set vertex expansion is optimal among polynomial-time certificates up to leading order as $\varepsilon \to 0$.

However, they note a remaining gap exactly at the critical value $d/2$ for certain degrees $d=q+1$ and leave open whether certifying this exact bound is feasible or hard. Resolving this would sharpen the frontier between achievable spectral certification and conjectured hardness at the threshold.