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

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 $ε$-DP algorithm with additive error $\tilde{O}(n^{2/3} / ε)$ and an $(ε, δ)$-DP algorithm with additive error $\tilde{O}(\sqrt{n} / ε)$ 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 $Ω(n^{1/6})$ is necessary for any sufficiently small $ε, δ> 0$. 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 $\tilde{O}\left(n^{1/2 + O(1/k)} / ε\right)$ in the $ε$-DP case and $\tilde{O}(n^{1/3 + O(1/k)} / ε)$ in the $(ε, δ)$-DP case, respectively.