It is shown in this note that approximating the number of independent sets in a $k$-uniform linear hypergraph with maximum degree at most $\Delta$ is NP-hard if $\Delta\geq 5\cdot 2^{k-1}+1$. 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).

翻译：本文证明了在最大度至多为 $\Delta$ 的 $k$-均匀线性超图中，近似计算独立集的数量在 $\Delta\geq 5\cdot 2^{k-1}+1$ 时是NP-难的。这一结果确认了在相关采样和近似计数问题中，当前最优算法所适用的最大度范围（除去一些细微因子）是紧的。这些算法包括：Hermon、Sly 和 Zhang (RSA, 2019) 的近似采样器和随机近似方案，Qiu、Wang 和 Zhang (ICALP, 2022) 的完美采样器，以及 Feng、Guo、Wang、Wang 和 Yin (FOCS, 2023) 的确定性近似方案。