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（RIGs）中高效恢复团结构的算法。在此模型中，通过以概率$δ$独立选择顶点参与每个团，随机植入了$d = n^{Ω(1)}$个规模约为$k$的团。尽管已有大量工作致力于在随机图中恢复单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鲜有探索。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形（如Kothari、Vempala、Wein和Xu近期在[COLT'23]中的研究）更具挑战性。本工作首次提出了从精确恢复与近似恢复两个角度高效恢复RIG社区结构的算法。我们的算法进一步具备对噪声、单调对抗性攻击及特定最优数量边破坏的鲁棒性。这些算法在$k \gg \sqrt{n \log(n)}$条件下均有效。我们的技术遵循基于平方和层次结构的证明到算法框架。