One of the most important recent developments in the complexity of approximate counting is the classification of the complexity of approximating the partition functions of antiferromagnetic 2-spin systems on bounded-degree graphs. This classification is based on a beautiful connection to the so-called uniqueness phase transition from statistical physics on the infinite Delta-regular tree.  In this talk we study the impact of this classification on unweighted 2-spin models on k-uniform hypergraphs. As has already been indicated by Yin and Zhao, the connection between the uniqueness phase transition  and the complexity of approximate counting  breaks down in the hypergraph setting.

 

Nevertheless, we show that for every non-trivial symmetric k-ary Boolean function  f there exists a degree bound Delta_0 so that for all Delta >= Delta_0 the following problem is NP-hard: given a k-uniform hypergraph with maximum degree at most Delta,  approximate the partition function of the hypergraph 2-spin model associated with f. It is NP-hard to approximate this partition function even within an exponential factor. By contrast, if f is a trivial symmetric Boolean function (e.g., any function f that is excluded from our result), then the partition function of the corresponding hypergraph 2-spin model can be computed exactly in polynomial time. (joint work with Andreas Galanis)