We propose an efficient $\epsilon$-differentially private algorithm, that given a simple {\em weighted} $n$-vertex, $m$-edge graph $G$ with a \emph{maximum unweighted} degree $\Delta(G) \leq n-1$, outputs a synthetic graph which approximates the spectrum with $\widetilde{O}(\min\{\Delta(G), \sqrt{n}\})$ bound on the purely additive error. To the best of our knowledge, this is the first $\epsilon$-differentially private algorithm with a non-trivial additive error for approximating the spectrum of the graph. One of the subroutines of our algorithm also precisely simulates the exponential mechanism over a non-convex set, which could be of independent interest given the recent interest in sampling from a {\em log-concave distribution} defined over a convex set. Spectral approximation also allows us to approximate all possible $(S,T)$-cuts, but it incurs an error that depends on the maximum degree, $\Delta(G)$. We further show that using our sampler, we can also output a synthetic graph that approximates the sizes of all $(S,T)$-cuts on $n$ vertices weighted graph $G$ with $m$ edges while preserving $(\epsilon,\delta)$-differential privacy and an additive error of $\widetilde{O}(\sqrt{mn}/\epsilon)$. We also give a matching lower bound (with respect to all the parameters) on the private cut approximation for weighted graphs. This removes the gap of $\sqrt{W_{\mathsf{avg}}}$ in the upper and lower bound in Eli{\'a}{\v{s}}, Kapralov, Kulkarni, and Lee (SODA 2020), where $W_{\mathsf{avg}}$ is the average edge weight.

翻译：我们提出了一种高效的ε-差分隐私算法，该算法输入一个简单的加权n顶点、m边图G（其最大无权度Δ(G) ≤ n-1），输出一个合成图，其谱逼近的纯加性误差为Õ(min{Δ(G), √n})。据我们所知，这是首个在谱逼近中实现非平凡加性误差的ε-差分隐私算法。我们算法中的一个子程序能在非凸集上精确模拟指数机制，考虑到近期对凸集上对数凹分布采样的研究兴趣，此子程序可能具有独立价值。谱逼近还使我们能够逼近所有(S,T)-割，但其误差依赖于最大度Δ(G)。我们进一步证明，使用该采样器，我们还能输出一个合成图，在保留(ε,δ)-差分隐私且加性误差为Õ(√mn/ε)的前提下，逼近所有含n顶点、m边的加权图G中(S,T)-割的大小。我们还给出了加权图私有割逼近的下界（针对所有参数），这消除了Eliáš、Kapralov、Kulkarni和Lee（SODA 2020）中上下界之间的√W_avg差距，其中W_avg为平均边权。