We study the algorithmic task of finding a large independent set in a sparse Erd\H{o}s-R\'{e}nyi random graph with $n$ vertices and average degree $d$. The maximum independent set is known to have size $(2 \log d / d)n$ in the double limit $n \to \infty$ followed by $d \to \infty$, but the best known polynomial-time algorithms can only find an independent set of half-optimal size $(\log d / d)n$. We show that the class of low-degree polynomial algorithms can find independent sets of half-optimal size but no larger, improving upon a result of Gamarnik, Jagannath, and the author. This generalizes earlier work by Rahman and Vir\'ag, which proved the analogous result for the weaker class of local algorithms.

中文翻译：

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最大独立集的最优低度硬度

我们研究了在具有 $n$ 个顶点和平均度数 $d$ 的稀疏 Erd\H{o}sR\'{e}nyi 随机图中找到一个大的独立集的算法任务。已知最大独立集的大小为 $(2 \log d / d)n$ 在双极限 $n \to \infty$ 后跟 $d \to \infty$，但最著名的多项式时间算法可以只找到一组独立的半最优大小 $(\log d / d)n$。我们表明，低次多项式算法类可以找到半最优大小但不能更大的独立集合，这改进了 Gamarnik、Jagannath 和作者的结果。这概括了 Rahman 和 Vir\'ag 的早期工作，后者证明了较弱的局部算法类的类似结果。