Differentially Private Gomory-Hu Trees
Anders Aamand, Justin Y. Chen, Mina Dalirrooyfard, Slobodan Mitrović, Yuriy Nevmyvaka, Sandeep Silwal, Yinzhan Xu
TL;DR
The paper tackles constructing a differentially private all-pairs min-cut representation by designing an ε-DP algorithm that outputs an approximate Gomory-Hu tree for weighted undirected graphs. The approach hinges on private Min Isolating Cuts and a carefully controlled recursive GH-tree construction that maintains privacy across polylogarithmic-depth recursion while achieving near-optimal additive error tilde O(n/ε). The main contributions are the DP Min Isolating Cuts with provable error guarantees, a private Gomory-Hu tree construction with an overall additive error of tilde O(n/ε) per pair, and corollaries including a polynomial-time pure-DP algorithm for global Min-Cut and implications for minimum k-cut, all within near-quadratic runtime. These results close gaps between private and non-private cut problems by showing the additive error required to privatize APMC is closer to Min-s-t-Cut than to all-cuts, enabling useful private cut information while preserving data privacy in graph analyses.
Abstract
Given an undirected, weighted $n$-vertex graph $G = (V, E, w)$, a Gomory-Hu tree $T$ is a weighted tree on $V$ such that for any pair of distinct vertices $s, t \in V$, the Min-$s$-$t$-Cut on $T$ is also a Min-$s$-$t$-Cut on $G$. Computing a Gomory-Hu tree is a well-studied problem in graph algorithms and has received considerable attention. In particular, a long line of work recently culminated in constructing a Gomory-Hu tree in almost linear time [Abboud, Li, Panigrahi and Saranurak, FOCS 2023]. We design a differentially private (DP) algorithm that computes an approximate Gomory-Hu tree. Our algorithm is $\varepsilon$-DP, runs in polynomial time, and can be used to compute $s$-$t$ cuts that are $\tilde{O}(n/\varepsilon)$-additive approximations of the Min-$s$-$t$-Cuts in $G$ for all distinct $s, t \in V$ with high probability. Our error bound is essentially optimal, as [Dalirrooyfard, Mitrović and Nevmyvaka, NeurIPS 2023] showed that privately outputting a single Min-$s$-$t$-Cut requires $Ω(n)$ additive error even with $(1, 0.1)$-DP and allowing for a multiplicative error term. Prior to our work, the best additive error bounds for approximate all-pairs Min-$s$-$t$-Cuts were $O(n^{3/2}/\varepsilon)$ for $\varepsilon$-DP [Gupta, Roth and Ullman, TCC 2012] and $O(\sqrt{mn} \cdot \text{polylog}(n/δ) / \varepsilon)$ for $(\varepsilon, δ)$-DP [Liu, Upadhyay and Zou, SODA 2024], both of which are implied by differential private algorithms that preserve all cuts in the graph. An important technical ingredient of our main result is an $\varepsilon$-DP algorithm for computing minimum Isolating Cuts with $\tilde{O}(n / \varepsilon)$ additive error, which may be of independent interest.