Finding min $s$-$t$ cuts in graphs is a basic algorithmic tool with applications in image segmentation, community detection, reinforcement learning, and data clustering. In this problem, we are given two nodes as terminals, and the goal is to remove the smallest number of edges from the graph so that these two terminals are disconnected. We study the complexity of differential privacy for the min $s$-$t$ cut problem and show nearly tight lower and upper bounds where we achieve privacy at no cost for running time efficiency. We also develop a differentially private algorithm for the multiway $k$-cut problem, in which we are given $k$ nodes as terminals that we would like to disconnect. As a function of $k$, we obtain privacy guarantees that are exponentially more efficient than applying the advanced composition theorem to known algorithms for multiway $k$-cut. Finally, we empirically evaluate the approximation of our differentially private min $s$-$t$ cut algorithm and show that it almost matches the quality of the output of non-private ones.

翻译：在图论中寻找最小$s$-$t$割是一种基础算法工具，广泛应用于图像分割、社区检测、强化学习和数据聚类。该问题中，给定两个节点作为终端，目标是从图中移除最少边数使得这两个终端不连通。我们研究了最小$s$-$t$割问题的差分隐私复杂度，并给出了近乎紧的下界与上界——在实现隐私保护的同时不增加运行时间开销。此外，我们针对多路$k$割问题（给定$k$个终端节点需使其互不连通）设计了差分隐私算法。实验结果表明，该算法能够以与$k$相关的隐私保证，其效率相比将高级组合定理直接应用于已知多路$k$割算法呈指数级提升。最后，我们通过实证评估验证了所提差分隐私最小$s$-$t$割算法的近似性能，显示其输出质量几乎媲美非隐私算法。