Differentially Private All-Pairs Min Cut Values

Abstract

We study the problem of privately releasing the min cut values of all pairs of vertices in a given $n$-node weighted graph, under the edge-neighboring model. Straightforward application of advanced composition yields an approximate DP algorithm that outputs all min cut values up to $\tilde{O}(n)$ additive error. Prior works [Dalirrooyfard, Mitrovi{ć} and Nevmyvaka, 2023; Aamand et al., 2026] have established that $\tildeΘ(n)$ is the correct bound if the algorithm also needs to output cuts that realize the approximate min cut values. On the other hand, simply outputting the values could be an easier problem, but prior algorithms have been stuck on the $\tilde{O}(n)$ bound. In this work, we give an approximate DP algorithm outputting all-pair min cut values with $\tilde{O}(\sqrt{n})$ additive error, beating the linear error barrier. Our algorithm applies the sparse vector technique [Dwork and Roth, 2014] to maintain a hierarchical decomposition of the vertices based on the min cut structure. We also prove an $Ω(\log^{3/2} n)$ lower bound for this problem.