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：本文研究在稠密随机交集图（RIG）中恢复团结构的高效算法。在此模型中，通过以概率δ独立选择顶点参与每个团，随机植入了约d = n^{Ω(1)}个规模近似为k的团。尽管在随机图中恢复单个或多个不相交植入团的研究已有大量成果，但将此问题自然扩展至恢复重叠团的任务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恢复RIG社区结构的有效算法。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 ≫ √(n log n)时均适用，其技术路线遵循基于平方和分层体系的证明到算法框架。