Computing the core decomposition of a graph is a fundamental problem that has recently been studied in the differentially private setting, motivated by practical applications in data mining. In particular, Dhulipala et al. [FOCS 2022] gave the first mechanism for approximate core decomposition in the challenging and practically relevant setting of local differential privacy. One of the main open problems left by their work is whether the accuracy, i.e., the approximation ratio and additive error, of their mechanism can be improved. We show the first lower bounds on the additive error of approximate and exact core decomposition mechanisms in the centralized and local model of differential privacy, respectively. We also give mechanisms for exact and approximate core decomposition in the local model, with almost matching additive error bounds. Our mechanisms are based on a black-box application of continual counting. They also yield improved mechanisms for the approximate densest subgraph problem in the local model.

翻译：计算图的核心分解是一个基础问题，近年来在差分隐私设置下受到关注，其动机源于数据挖掘中的实际应用。具体而言，Dhulipala 等人 [FOCS 2022] 在具有挑战性且实际相关的局部差分隐私设置中，首次提出了近似核心分解的机制。他们工作遗留的主要开放问题之一在于能否提升该机制的精度（即近似比和加性误差）。我们分别针对集中式模型和局部模型下的近似及精确核心分解机制，首次给出了加性误差的下界。此外，我们提出了局部模型下精确与近似核心分解的机制，其加性误差界几乎匹配。这些机制基于连续计数的黑箱应用，同时改进了局部模型下近似最密子图问题的现有机制。