We study efficient algorithms for recovering cliques in dense random intersection graphs (RIGs). In this model, $d = n^{Ω(1)}$ cliques of size approximately $k$ are randomly planted by choosing the vertices to participate in each clique independently with probability $δ$. While there has been extensive work on recovering one, or multiple disjointly planted cliques in random graphs, the natural extension of this question to recovering overlapping cliques has been, surprisingly, largely unexplored. Moreover, because every vertex can be part of polynomially many cliques, this task is significantly more challenging than in case of disjointly planted cliques (as recently studied by Kothari, Vempala, Wein and Xu [COLT'23]). In this work we obtain the first efficient algorithms for recovering the community structure of RIGs both from the perspective of exact and approximate recovery. Our algorithms are further robust to noise, monotone adversaries, and a certain, optimal number of edge corruptions. They work whenever $k \gg \sqrt{n \log(n)}$. Our techniques follow the proofs-to-algorithms framework utilizing the sum-of-squares hierarchy.

翻译：我们研究了在稠密随机交集图中恢复团的高效算法。在该模型中，通过以概率$δ$独立选择顶点参与每个团，随机植入了大约$k$个规模的$d = n^{Ω(1)}$个团。尽管已有大量工作致力于在随机图中恢复一个或多个不相交的植入团，但将此问题自然扩展至恢复重叠团的研究却惊人地未被充分探索。此外，由于每个顶点可能属于多项式数量的团，该任务比不相交植入团的情况（如Kothari、Vempala、Wein和Xu近期研究[COLT'23]）更具挑战性。本工作中，我们首次提出了从精确恢复和近似恢复两个角度恢复随机交集图社区结构的高效算法。我们的算法进一步对噪声、单调对抗性攻击以及特定最优数量的边破坏具有鲁棒性。这些算法在$k \gg \sqrt{n \log(n)}$条件下均有效。我们的技术遵循证明到算法的框架，并利用了平方和层次结构。