The complexity of approximating averages on bounded-degree graphs

We prove that, unless P=NP, there is no polynomial-time algorithm to approximate within some multiplicative constant the average size of an independent set in graphs of maximum degree 6. This is a special case of a more general result for the hard-core model defined on independent sets weighted by a parameter $\lambda>0$. In the general setting, we prove that, unless P=NP, for all $\Delta\geq 3$, all $\lambda>\lambda_c(\Delta)$, there is no FPTAS which applies to all graphs of maximum degree $\Delta$ for computing the average size of the independent set in the Gibbs distribution, where $\lambda_c(\Delta)$ is the critical point for the uniqueness/non-uniqueness phase transition on the $\Delta$-regular tree. Moreover, we prove that for $\lambda$ in a dense set of this non-uniqueness region the problem is NP-hard to approximate within some constant factor. Our work extends to the antiferromagnetic Ising model and generalizes to all 2-spin antiferromagnetic models, establishing hardness of computing the average magnetization in the tree non-uniqueness region. Previously, Schulman, Sinclair and Srivastava (2015) showed that it is #P-hard to compute the average magnetization exactly, but no hardness of approximation results were known. Hardness results of Sly (2010) and Sly and Sun (2014) for approximating the partition function do not imply hardness of computing averages. The new ingredient in our reduction is an intricate construction of pairs of rooted trees whose marginal distributions at the root agree but their derivatives disagree. The main technical contribution is controlling what marginal distributions and derivatives are achievable and using Cauchy's functional equation to argue existence of the gadgets.