The complexity of approximately counting in 2-spin systems on $k$-uniform bounded-degree hypergraphs

TL;DR

This paper investigates the computational complexity of approximately counting partition functions in hypergraph 2-spin models, revealing NP-hardness results for non-trivial functions and polynomial-time computability for trivial functions.

Contribution

It extends the complexity classification of 2-spin systems from graphs to hypergraphs, showing NP-hardness for non-trivial functions and tractability for trivial functions.

Findings

NP-hard to approximate partition functions for non-trivial symmetric functions on hypergraphs

Polynomial-time algorithms exist for trivial symmetric functions

Breakdown of the uniqueness phase transition connection in hypergraph settings

Abstract

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. Our objective is to 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 $\Delta \geq \Delta_0$ 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.