Hardness of detecting noisy lifts of Ramanujan graphs (formal conjecture)

Prove that for every fixed d-regular Ramanujan base multigraph H on k vertices and every sufficiently small noise level ε>0, no polynomial-time algorithm achieves strong detection between (i) the noisy lift distribution S^{rand}_ε L_m(H) and the uniformly random d-regular graph G(n,d), and likewise under any of the alternative noise models \widetilde{S}^{rand}_ε, S^{adv}_ε, or \widetilde{S}^{adv}_ε; and (ii) for every fixed bipartite d-regular Ramanujan base multigraph H, no polynomial-time algorithm achieves strong detection between the noisy lift distribution S^{rand,bi}_ε L_m(H) and the uniformly random bipartite d-regular graph G((n/2,n/2),d), and likewise under \widetilde{S}^{rand,bi}_ε, S^{adv}_ε, or \widetilde{S}^{adv}_ε. Here strong detection means both Type I and Type II errors tend to zero as m→∞.

Background

The paper studies the hypothesis-testing task of distinguishing random lifts of a fixed base graph H from uniformly random d-regular graphs, including bipartite variants. Because simple spectral certificates are defeated by small perturbations, the authors introduce noise models that change at most εn edges while preserving regularity and, for respectful variants, the fiber structure of the lift.

They formulate a precise computational hardness conjecture stating that when the base graph H is (bipartite) Ramanujan, no polynomial-time algorithm can strongly distinguish a noisy lift of H from a uniformly random (bipartite) d-regular graph. This formalizes and generalizes an informal conjecture stated earlier and underpins their conditional lower bounds for certifying various graph properties.