Recent inapproximability results of Sly (2010), together with an
approximation algorithm presented by Weitz (2006) establish a beautiful picture
for the computational complexity of approximating the partition function of the
hard-core model. Let λc​(TΔ​) denote the critical activity for the
hard-model on the infinite Δ-regular tree. Weitz presented an FPTAS for
the partition function when λ<λc​(TΔ​) for graphs with
constant maximum degree Δ. In contrast, Sly showed that for all
Δ≥3, there exists ϵΔ​>0 such that (unless RP=NP) there
is no FPRAS for approximating the partition function on graphs of maximum
degree Δ for activities λ satisfying
λc​(TΔ​)<λ<λc​(TΔ​)+ϵΔ​.
We prove that a similar phenomenon holds for the antiferromagnetic Ising
model. Recent results of Li et al. and Sinclair et al. extend Weitz's approach
to any 2-spin model, which includes the antiferromagnetic Ising model, to yield
an FPTAS for the partition function for all graphs of constant maximum degree
Δ when the parameters of the model lie in the uniqueness regime of the
infinite tree TΔ​. We prove the complementary result that for the
antiferrogmanetic Ising model without external field that, unless RP=NP, for
all Δ≥3, there is no FPRAS for approximating the partition function
on graphs of maximum degree Δ when the inverse temperature lies in the
non-uniqueness regime of the infinite tree TΔ​. Our results extend to a
region of the parameter space for general 2-spin models. Our proof works by
relating certain second moment calculations for random Δ-regular
bipartite graphs to the tree recursions used to establish the critical points
on the infinite tree.Comment: Journal version (no changes