anti-ferromagnetic: By<1 bounded△or△=∞ (B,y,A)lies in the interiors of uniqueness regions of d-regular trees for all ds A. 3 FPTAS for graphs of max-degree A [Sly-Sun'12] [Galanis-Stefankovic-Vigoda'12]: (B,y,A)lies in the interiors of non-uniqueness regions of a-regular trees for some d≤△. NR assuming FPRAS for graphs of max-degree Aanti-ferromagnetic: ￾￾ < 1 ∃ FPTAS for graphs of max-degree Δ (β, γ, λ) lies in the interiors of uniqueness regions of d-regular trees for all d ≤ Δ. ∄ FPRAS for graphs of max-degree Δ (β, γ, λ) lies in the interiors of non-uniqueness regions of d-regular trees for some d ≤ Δ. assuming NP ≠RP [Sly-Sun’12] [Galanis-Stefankovic-Vigoda’12]: bounded Δ or Δ=∞