对超图的独立集性质进行测试
Testing the Independent Set Property in Hypergraphs
AI总结：
本文提出了对超图独立集性质进行测试的样本复杂度新上界，并通过超图容器方法实现了对q依赖性的指数级改进。
AI中文摘要：
对具有n个顶点的图是否包含大小为ρn的独立集，或是否与具有大小为ρn的独立集的图在ε意义下相距甚远的最优样本复杂度已被确定为~O(ρ³/ε²)，这是Blais和Seth（SICOMP 2025）的一项重要成果。相比之下，在q-均匀超图中，已知的上界和下界之间存在显著差距，且在过去二十年中该问题几乎没有进展。在本文中，我们证明了测试ρ-独立集性质的样本复杂度的新上界为~O(qρ^{2q-3}/(ε²(q-2)!²))。之前的最佳上界为~O(2^q q! ρ^{2q}/ε³)，由Langberg（RANDOM 2004）提出。该结果在ε的依赖性上是最佳的，并在q的依赖性上实现了指数级的改进。我们通过超图容器方法的新应用来证明这一结果。
英文摘要：
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.
