In this paper, we address the challenge of differential privacy in the context of graph cuts, specifically focusing on the multiway cut and the minimum $k$-cut. We introduce edge-differentially private algorithms that achieve nearly optimal performance for these problems. Motivated by multiway cut, we propose the shifting mechanism, a general framework for private combinatorial optimization problems. This framework allows us to develop an efficient private algorithm with a multiplicative approximation ratio that matches the state-of-the-art non-private algorithm, improving over previous private algorithms that have provably worse multiplicative loss. We then provide a tight information-theoretic lower bound on the additive error, demonstrating that for constant $k$, our algorithm is optimal in terms of the privacy cost. The shifting mechanism also allows us to design private algorithm for the multicut and max-cut problems, with runtimes determined by the best non-private algorithms for these tasks. For the minimum $k$-cut problem we use a different approach, combining the exponential mechanism with bounds on the number of approximate $k$-cuts to get the first private algorithm with optimal additive error of $O(k\log n)$ (for a fixed privacy parameter). We also establish an information-theoretic lower bound that matches this additive error. Furthermore, we provide an efficient private algorithm even for non-constant $k$, including a polynomial-time 2-approximation with an additive error of $\tilde{O}(k^{1.5})$.

翻译：本文针对图割问题中的差分隐私挑战展开研究，重点探讨多路割与最小$k$-割问题。我们提出了边差分隐私算法，在这些问题上实现了近乎最优的性能。受多路割问题启发，我们提出了偏移机制——一个适用于隐私保护组合优化问题的通用框架。该框架使我们能够开发出具有乘法近似比的高效隐私算法，其近似比与最先进的非隐私算法相当，较之先前具有可证明更大乘法损失的隐私算法有所改进。我们进一步给出了关于加法误差的紧信息论下界，证明在常数$k$的情况下，我们的算法在隐私代价方面达到最优。偏移机制还使我们能够为多割与最大割问题设计隐私算法，其运行时间由这些任务的最佳非隐私算法决定。针对最小$k$-割问题，我们采用不同方法，将指数机制与近似$k$-割数量的界限相结合，首次实现了具有$O(k\log n)$最优加法误差的隐私算法（针对固定隐私参数）。我们还建立了与该加法误差匹配的信息论下界。此外，我们为非恒定$k$的情况提供了高效的隐私算法，包括具有$\tilde{O}(k^{1.5})$加法误差的多项式时间2-近似算法。