Differential Privacy is the gold standard in privacy-preserving data analysis. This paper addresses the challenge of producing a differentially edge-private vertex coloring. In this paper, we present two novel algorithms to approach this problem. Both algorithms initially randomly colors each vertex from a fixed size palette, then applies the exponential mechanism to locally resample colors for either all or a chosen subset of the vertices. Any non-trivial differentially edge private coloring of graph needs to be defective. A coloring of a graph is k defective if all vertices of the graph share it's assigned color with at most k of its neighbors. This is the metric by which we will measure the utility of our algorithms. Our first algorithm applies to d-inductive graphs. Assume we have a d-inductive graph with n vertices and max degree $Δ$. We show that our algorithm provides a \(3ε\)-differentially private coloring with \(O(\frac{\log n}ε+d)\) max defectiveness, given a palette of size $Θ(\fracΔ{\log n}+\frac{1}ε)$ Furthermore, we show that this algorithm can generalize to $O(\fracΔ{cε}+d)$ defectiveness, where c is the size of the palette and $c=O(\fracΔ{\log n})$. Our second algorithm utilizes noisy thresholding to guarantee \(O(\frac{\log n}ε)\) max defectiveness, given a palette of size $Θ(\fracΔ{\log n}+\frac{1}ε)$, generalizing to all graphs rather than just d-inductive ones.

翻译：差分隐私是隐私保护数据分析领域的黄金标准。本文致力于解决生成差分边隐私顶点着色的挑战。我们提出了两种新颖算法来解决该问题。两种算法首先从固定大小的调色板中为每个顶点随机着色，然后应用指数机制对所有顶点或选定顶点子集进行局部颜色重采样。任何非平凡的差分边隐私图着色都必须是缺陷着色。若图中所有顶点与其至多k个邻居共享所分配颜色，则称该着色为k缺陷着色。我们将以此度量标准评估算法的效用。第一种算法适用于d归纳图。假设我们有一个包含n个顶点、最大度为$Δ$的d归纳图。我们证明在调色板大小为$Θ(\fracΔ{\log n}+\frac{1}ε)$的条件下，该算法能提供具有\(O(\frac{\log n}ε+d)\)最大缺陷度的\(3ε\)-差分隐私着色。此外，我们证明该算法可推广至$O(\fracΔ{cε}+d)$缺陷度，其中c为调色板尺寸且满足$c=O(\fracΔ{\log n})$。第二种算法利用噪声阈值技术，在调色板大小为$Θ(\fracΔ{\log n}+\frac{1}ε)$的条件下，保证\(O(\frac{\log n}ε)\)的最大缺陷度，且可推广至所有图类而不仅限于d归纳图。