We present the notion of \emph{reasonable utility} for binary mechanisms, which applies to all utility functions in the literature. This notion induces a partial ordering on the performance of all binary differentially private (DP) mechanisms. DP mechanisms that are maximal elements of this ordering are optimal DP mechanisms for every reasonable utility. By looking at differential privacy as a randomized graph coloring, we characterize these optimal DP in terms of their behavior on a certain subset of the boundary datasets we call a boundary hitting set. In the process of establishing our results, we also introduce a useful notion that generalizes DP conditions for binary-valued queries, which we coin as suitable pairs. Suitable pairs abstract away the algebraic roles of $\varepsilon,\delta$ in the DP framework, making the derivations and understanding of our proofs simpler. Additionally, the notion of a suitable pair can potentially capture privacy conditions in frameworks other than DP and may be of independent interest.

翻译：我们提出了二元机制的“合理效用”概念，该概念适用于文献中的所有效用函数。这一概念对二元差分隐私（DP）机制的性能引入了偏序关系。该偏序中的极大元即对于所有合理效用而言的最优DP机制。通过将差分隐私视为随机图着色，我们根据这些最优DP机制在特定边界数据集子集（称为边界命中集）上的行为对其进行了刻画。在建立结果的过程中，我们还引入了一个有用概念，即二元值查询的DP条件推广形式，我们称之为“恰当对”。恰当对抽象出了DP框架中$\varepsilon,\delta$的代数角色，简化了推导过程与对证明的理解。此外，恰当对的概念可能捕捉到DP之外其他框架中的隐私条件，并可能具有独立的研究价值。