Testing the Independent Set Property in Hypergraphs

2026-07-14

AI Summary

This work addresses the sample complexity of testing independence in q-uniform hypergraphs, aiming to close a long-standing gap between known upper and lower bounds. By introducing the hypergraph container method—an approach previously unused in this context—and combining it with probabilistic combinatorial analysis, the authors design an efficient property tester that distinguishes whether a hypergraph contains an independent set of size ρn or is ε-far from having one. The proposed method achieves optimal dependence on ε and yields an exponential improvement in its dependence on q, reducing the sample complexity upper bound from Õ(2^q q! ρ^{2q}/ε³) to Õ(q ρ^{2q−3}/(ε² (q−2)!²)).

Abstract

The optimal sample complexity of testing if an $n$-vertex graph has an independent set of size $ρn$, or is $\varepsilon$-far from having an independent set of size $ρn$, was established to be $\widetilde{O}(ρ^3/\varepsilon^2)$, in a notable result by Blais and Seth (SICOMP 2025). In contrast, for $q$-uniform hypergraphs, there is a significant gap between the best known upper and lower bounds, and there has been no progress on the problem for the last two decades. In this work, we prove a new upper bound of $\widetilde{O}\!\left(\frac{qρ^{2q-3}}{\varepsilon^2 (q-2)!^2}\right)$ on the sample complexity of testing the $ρ$-independent set property. The previous best known upper bound was $\widetilde{O}\!\left(\frac{2^q q! ρ^{2q}}{\varepsilon^3}\right)$, due to Langberg (RANDOM 2004). This establishes the optimal dependence on $\varepsilon$ and gives an exponential improvement in the dependence on $q$. We prove our result via a new application of the hypergraph container method.

Problem

• hypergraph
• independent set
• property testing
• sample complexity

Innovation

• hypergraph container method
• property testing
• independent set
• sample complexity
• uniform hypergraphs