In this paper, we address the challenge of differential privacy in the context of graph cuts, specifically focusing on the minimum $k$-cut and multiway cut problems. We introduce edge-differentially private algorithms that achieve nearly optimal performance for these problems. For the multiway cut problem, we first provide a private algorithm with a multiplicative approximation ratio that matches the state-of-the-art non-private algorithm. We then present a tight information-theoretic lower bound on the additive error, demonstrating that our algorithm on weighted graphs is near-optimal for constant $k$. For the minimum $k$-cut problem, our algorithms leverage a known bound on the number of approximate $k$-cuts, resulting in a private algorithm with optimal additive error $O(k\log n)$ for fixed privacy parameter. We also establish a information-theoretic lower bound that matches this additive error. Additionally, we give an efficient private algorithm for $k$-cut even for non-constant $k$, including a polynomial-time 2-approximation with an additive error of $\widetilde{O}(k^{1.5})$.

翻译：本文针对图割问题中的差分隐私挑战展开研究，重点探讨最小$k$割与多路割问题。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。对于多路割问题，我们首先提出了一种具有乘法近似比的隐私算法，其近似比与最先进的非隐私算法相匹配。随后我们通过严格的信息论下界证明了加性误差的理论极限，表明在加权图上我们的算法对于常数$k$是近乎最优的。对于最小$k$割问题，我们的算法利用已知的近似$k$割数量界限，在固定隐私参数条件下实现了最优加性误差$O(k\log n)$的隐私算法。我们还建立了与该加性误差相匹配的信息论下界。此外，针对非常数$k$的情况，我们提出了高效的$k$割隐私算法，包括具有$\widetilde{O}(k^{1.5})$加性误差的多项式时间2-近似算法。