Inapproximability of Counting Independent Sets in Linear Hypergraphs

It is shown in this note that approximating the number of independent sets in a $k$-uniform linear hypergraph with maximum degree at most $\mathrm{\Delta }$ is NP-hard if $\mathrm{\Delta }\ge 5\cdot 2^{k-1}+1$. This confirms that for the relevant sampling and approximate counting problems, the regimes on the maximum degree where the state-of-the-art algorithms work are tight, up to some small factors. These algorithms include: the approximate sampler and randomised approximation scheme by Hermon, Sly and Zhang (RSA, 2019), the perfect sampler by Qiu, Wang and Zhang (ICALP, 2022), and the deterministic approximation scheme by Feng, Guo, Wang, Wang and Yin (FOCS, 2023).