We study the following combinatorial counting and sampling problems: can we efficiently sample from the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ conditioned on triangle-freeness? Can we efficiently approximate the probability that $G(n,p)$ is triangle-free? These are prototypical instances of forbidden substructure problems ubiquitous in combinatorics. The algorithmic questions are instances of approximate counting and sampling for a hypergraph hard-core model. Estimating the probability that $G(n,p)$ has no triangles is a fundamental question in probabilistic combinatorics and one that has led to the development of many important tools in the field. Through the work of several authors, the asymptotics of the logarithm of this probability are known if $p =o( n^{-1/2})$ or if $p =\omega( n^{-1/2})$. The regime $p = \Theta(n^{-1/2})$ is more mysterious, as this range witnesses a dramatic change in the the typical structural properties of $G(n,p)$ conditioned on triangle-freeness. As we show, this change in structure has a profound impact on the performance of sampling algorithms. We give two different efficient sampling algorithms for triangle-free graphs (and complementary algorithms to approximate the triangle-freeness large deviation probability), one that is efficient when $p < c/\sqrt{n}$ and one that is efficient when $p > C/\sqrt{n}$ for constants $c, C>0$. The latter algorithm involves a new approach for dealing with large defects in the setting of sampling from low-temperature spin models.

翻译：我们研究以下组合计数与抽样问题：能否从Erdős-Rényi随机图$G(n,p)$在给定无三角形条件下高效抽样？能否高效逼近$G(n,p)$为无三角形图的概率？这些是组合学中普遍存在的禁止子结构问题的典型实例。该算法问题属于超图硬核模型的近似计数与抽样范畴。估计$G(n,p)$不含三角形的概率是概率组合学的基本问题，该问题推动了该领域许多重要工具的发展。通过多位学者的工作，当$p=o(n^{-1/2})$或$p=\omega(n^{-1/2})$时，该概率对数的渐近性质已明确。$p=\Theta(n^{-1/2})$区域则更为复杂，因为该区域见证了在无三角形条件下$G(n,p)$典型结构性质的剧烈变化。如我们所示，这种结构变化对抽样算法的性能产生深远影响。我们提出两种不同的无三角形图高效抽样算法（及配套的近似无三角形大偏差概率算法），一种在$p<c/\sqrt{n}$时高效，另一种在$p>C/\sqrt{n}$时高效（其中$c,C>0$为常数）。后一种算法提出了一种处理低温自旋模型抽样中大缺陷的新方法。