We give nearly optimal bounds on the sample complexity of $(\widetilde{\Omega}(\epsilon),\epsilon)$-tolerant testing the $\rho$-independent set property in the dense graph setting. In particular, we give an algorithm that inspects a random subgraph on $\widetilde{O}(\rho^3/\epsilon^2)$ vertices and, for some constant $c,$ distinguishes between graphs that have an induced subgraph of size $\rho n$ with fewer than $\frac{\epsilon}{c \log^4(1/\epsilon)} n^2$ edges from graphs for which every induced subgraph of size $\rho n$ has at least $\epsilon n^2$ edges. Our sample complexity bound matches, up to logarithmic factors, the recent upper bound by Blais and Seth (2023) for the non-tolerant testing problem, which is known to be optimal for the non-tolerant testing problem based on a lower bound by Feige, Langberg and Schechtman (2004). Our main technique is a new graph container lemma for sparse subgraphs instead of independent sets. We also show that our new lemma can be used to generalize one of the classic applications of the container method, that of counting independent sets in regular graphs, to counting sparse subgraphs in regular graphs.

翻译：我们针对稠密图设定中$(\widetilde{\Omega}(\epsilon),\epsilon)$-容错测试$\rho$-独立集性质的问题，给出了近乎最优的样本复杂度界限。具体而言，我们提出了一种算法，该算法检查一个包含$\widetilde{O}(\rho^3/\epsilon^2)$个顶点的随机子图，并且对于某个常数$c$，能够区分以下两类图：一类图存在一个大小为$\rho n$的诱导子图，其边数少于$\frac{\epsilon}{c \log^4(1/\epsilon)} n^2$；另一类图的每个大小为$\rho n$的诱导子图都至少有$\epsilon n^2$条边。我们的样本复杂度界限，在对数因子范围内，与Blais和Seth（2023）针对非容错测试问题最近给出的上界相匹配，而根据Feige、Langberg和Schechtman（2004）的下界，该上界对于非容错测试问题已知是最优的。我们的主要技术是一种新的、针对稀疏子图（而非独立集）的图容器引理。我们还展示了我们的新引理可用于将容器方法的经典应用之一——即计算正则图中的独立集数量——推广到计算正则图中的稀疏子图数量。