#BIS-Hardness for 2-Spin Systems on Bipartite Bounded Degree Graphs in the Tree Non-uniqueness Region

Abstract

Counting independent sets on bipartite graphs (#BIS) is considered a canonical counting problem of intermediate approximation complexity. It is conjectured that #BIS neither has an FPRAS nor is as hard as #SAT to approximate. We study #BIS in the general framework of two-state spin systems in bipartite graphs. Such a system is parameterized by three numbers (beta,gamma,lambda), where beta (respectively gamma) represents the weight of an edge (or "interaction strength") whose endpoints are of the same 0 (respectively 1) spin, and lambda is the weight of a 1 vertex, also known as an "external field". By convention, the edge weight with unequal 0/1 end points and the vertex weight with spin 0 are both normalized to 1. The partition function of the special case beta=1, gamma=0, and lambda=1 counts the number of independent sets. We define two notions, nearly-independent phase-correlated spins and symmetry breaking. We prove that it is #BIS-hard to approximate the partition function of any two-spin system on bipartite graphs supporting these two notions.

As a consequence, we show that #BIS on graphs of degree at most 6 is as hard to approximate as #BIS~without degree bound. The degree bound 6 is the best possible as Weitz presented an FPTAS to count independent sets on graphs of maximum degree 5. This result extends to the hard-core model and to other anti-ferromagnetic two-spin models. In particular, for all antiferromagnetic two-spin systems, namely those satisfying beta*gamma<1, we prove that when the infinite (Delta-1)-ary tree lies in the non-uniqueness region then it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree Delta, except for the case beta=gamma and lambda=1. The exceptional case is precisely the antiferromagnetic Ising model without an external field, and we show that it  has an FPRAS on bipartite graphs. Our inapproximability results match the approximability results of Li et al., who presented an FPTAS for general graphs of maximum degree Delta when the parameters lie in the uniqueness region.