Tight bounds for larger approximate-DP failure probabilities

Establish tight error bounds for differentially private minimum spanning tree, minimum-weight perfect matching, and hierarchical clustering algorithms under the edge-weight model with larger values of the approximate-differential-privacy parameter \(\delta\) than the polynomially small regime considered in the paper.

Background

The paper proves lower bounds for approximate differential privacy when $\delta$ is polynomially small, including bounds of order $\Omega(n\log(m/n)/\varepsilon)$ for worst-case minimum spanning tree and minimum-weight perfect matching instances and topology-dependent bounds for minimum spanning tree and hierarchical clustering. The authors note that the assumption $\delta < (1/n)^{\Omega(1)}$ is standard because mechanisms can release a $\delta$-fraction of a dataset while satisfying approximate differential privacy.

The unresolved issue is whether matching upper and lower error bounds can be obtained when $\delta$ is larger than this polynomially small regime. This concerns the limitations of reconstruction-based lower bounds and the extent to which approximate differential privacy can improve private graph optimization accuracy.