Inapproximability of counting independent sets in linear hypergraphs

Abstract

It is shown in this note that approximating the number of independent sets in a k-uniform linear hypergraph with maximum degree at most Δ is NP-hard if Δ≥5⋅2k−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 et al. (2019) [5], the perfect sampler by Qiu et al. (2022) [6], and the deterministic approximation scheme by Feng et al. (2023) [7].