Inapproximability of Counting Independent Sets in Linear Hypergraphs

TL;DR

This paper proves that approximating the number of independent sets in certain linear hypergraphs is NP-hard beyond a specific degree threshold, confirming the tightness of existing algorithms' regimes.

Contribution

It establishes NP-hardness for counting independent sets in linear hypergraphs with maximum degree above a certain threshold, clarifying the limits of current algorithms.

Findings

NP-hardness for degree ≥ 5·2^{k-1}+1

Tight bounds for sampling and approximation algorithms

Validation of the limits of existing algorithms

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 $\Delta$ 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).