Differentially Private All-Pairs Shortest Path Distances: Improved Algorithms and Lower Bounds

TL;DR

This paper presents improved differentially private algorithms for releasing all-pairs shortest path distances in weighted graphs, achieving lower additive errors and establishing lower bounds, thus advancing the understanding of privacy-utility trade-offs.

Contribution

It introduces new DP algorithms with reduced error bounds for all-pairs shortest path distances and proves lower bounds, answering open questions in the field.

Findings

P algorithm with ilde{O}(n^{2/3}/psilon) error

P algorithm with ilde{O}(\u221a{n}/psilon) error

Lower bound of pprox n^{1/6} error for any DP algorithm

Abstract

We study the problem of releasing the weights of all-pair shortest paths in a weighted undirected graph with differential privacy (DP). In this setting, the underlying graph is fixed and two graphs are neighbors if their edge weights differ by at most $1$ in the $\ell_1$-distance. We give an $\epsilon$-DP algorithm with additive error $\tilde{O}(n^{2/3} / \epsilon)$ and an $(\epsilon, \delta)$-DP algorithm with additive error $\tilde{O}(\sqrt{n} / \epsilon)$ where $n$ denotes the number of vertices. This positively answers a question of Sealfon (PODS'16), who asked whether a $o(n)$-error algorithm exists. We also show that an additive error of $\Omega(n^{1/6})$ is necessary for any sufficiently small $\epsilon, \delta > 0$. Finally, we consider a relaxed setting where a multiplicative approximation is allowed. We show that, with a multiplicative approximation factor $k$, %$2k - 1$, the additive error can be reduced to $\tilde{O}\left(n^{1/2 + O(1/k)} / \epsilon\right)$ in the $\epsilon$-DP case and $\tilde{O}(n^{1/3 + O(1/k)} / \epsilon)$ in the $(\epsilon, \delta)$-DP case, respectively.