Recent work by Dhulipala et al. \cite{DLRSSY22} initiated the study of the $k$-core decomposition problem under differential privacy via a connection between low round/depth distributed/parallel graph algorithms and private algorithms with small error bounds. They showed that one can output differentially private approximate $k$-core numbers, while only incurring a multiplicative error of $(2 +\eta)$ (for any constant $\eta >0$) and additive error of $\poly(\log(n))/\eps$. In this paper, we revisit this problem. Our main result is an $\eps$-edge differentially private algorithm for $k$-core decomposition which outputs the core numbers with no multiplicative error and $O(\text{log}(n)/\eps)$ additive error. This improves upon previous work by a factor of 2 in the multiplicative error, while giving near-optimal additive error. Our result relies on a novel generalized form of the sparse vector technique, which is especially well-suited for threshold-based graph algorithms; thus, we further strengthen the connection between distributed/parallel graph algorithms and differentially private algorithms.

翻译：Dhulipala等人近期的工作\cite{DLRSSY22}通过建立低轮次/深度分布式/并行图算法与具有小误差界的隐私算法之间的联系，首次研究了差分隐私下的k-核分解问题。他们证明可以输出差分隐私近似的k-核数，仅需承受$(2 +\eta)$的乘法误差（对任意常数$\eta >0$）和$\poly(\log(n))/\eps$的加法误差。在本文中，我们重新审视该问题。主要成果是提出一种$\eps$-边差分隐私的k-核分解算法，该算法能输出核数且无乘法误差，加法误差为$O(\text{log}(n)/\eps)$。这使得乘法误差较先前工作降低了2倍，同时实现了近最优的加法误差。我们的结果依赖于稀疏向量技术的一种新颖泛化形式，该形式特别适用于基于阈值的图算法；因此，我们进一步强化了分布式/并行图算法与差分隐私算法之间的联系。