Recent work by Dhulipala, Liu, Raskhodnikova, Shi, Shun, and Yu~\cite{DLRSSY22} initiated the study of the $k$-core decomposition problem under differential privacy. They show that approximate $k$-core numbers can be output while guaranteeing differential privacy, 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. With a little additional work, this implies improved algorithms for densest subgraph and low out-degree ordering under differential privacy. For low out-degree ordering, we give an $\eps$-edge differentially private algorithm which outputs an implicit orientation such that the out-degree of each vertex is at most $d+O(\log{n}/{\eps})$, where $d$ is the degeneracy of the graph. This improves upon the best known guarantees for the problem by a factor of $4$ and gives near-optimal additive error. For densest subgraph, we give an $\eps$-edge differentially private algorithm outputting a subset of nodes that induces a subgraph of density at least ${D^*}/{2}-O(\text{log}(n)/\eps)$, where $D^*$ is the density for the optimal subgraph.

翻译：最近，Dhulipala、Liu、Raskhodnikova、Shi、Shun和Yu的工作~\cite{DLRSSY22}开创了差分隐私下$k$-核分解问题的研究。他们表明，在保证差分隐私的同时，可以输出近似的$k$-核数，仅引入$(2 +\eta)$（对任意常数$\eta >0$）的乘法误差和$\poly(\log(n))/\eps$的加法误差。本文重新审视这一问题。我们的主要成果是：一种针对$k$-核分解的$\eps$-边差分隐私算法，该算法输出无乘法误差且加法误差为$O(\text{log}(n)/\eps)$的核数。相较于先前工作，该结果将乘法误差降低2倍，同时实现近最优的加法误差。通过额外工作，这进一步改进了差分隐私下最密子图问题和低出度排序问题的算法。对于低出度排序，我们提出一种$\eps$-边差分隐私算法，输出一个隐式定向，使得每个顶点的出度不超过$d+O(\log{n}/{\eps})$，其中$d$为图的退化度。该结果将该问题的最佳已知保证提升4倍，并给出近最优的加法误差。对于最密子图，我们给出一种$\eps$-边差分隐私算法，输出一个诱导子图密度至少为${D^*}/{2}-O(\text{log}(n)/\eps)$的节点子集，其中$D^*$为最优子图的密度。