Near-Universally-Optimal Differentially Private Minimum Spanning Trees
Richard Hladík, Jakub Tětek
TL;DR
This work addresses privately releasing a minimum spanning tree in a setting where the underlying unweighted graph is public but edge weights are private. It proves that a Laplace-noise MST release is universally near-optimal up to a $\mathcal{O}(\log n)$ factor for the $\ell_1$ neighbor relation and that the exponential mechanism, implemented in matrix-multiplication time $\mathcal{O}(n^\omega)$, achieves universal near-optimality for both $\ell_1$ and $\ell_\infty$ neighbors. Central to the results is a diameter parameter $D$ of the spanning-tree space and a reduction to constructing large sets of dissimilar spanning trees, enabling tight lower bounds via packing arguments and codes. The combination of these analyses yields near-tight, instance- and graph-specific optimality guarantees for private MST release, advancing the notion of universal optimality in differential privacy and suggesting broader applicability to other graph problems and privacy notions. The results have practical impact for releasing graph-derived structures under privacy constraints, providing principled, near-optimal procedures with provable guarantees and efficient implementations.
Abstract
Devising mechanisms with good beyond-worst-case input-dependent performance has been an important focus of differential privacy, with techniques such as smooth sensitivity, propose-test-release, or inverse sensitivity mechanism being developed to achieve this goal. This makes it very natural to use the notion of universal optimality in differential privacy. Universal optimality is a strong instance-specific optimality guarantee for problems on weighted graphs, which roughly states that for any fixed underlying (unweighted) graph, the algorithm is optimal in the worst-case sense, with respect to the possible setting of the edge weights. In this paper, we give the first such result in differential privacy. Namely, we prove that a simple differentially private mechanism for approximately releasing the minimum spanning tree is near-optimal in the sense of universal optimality for the $\ell_1$ neighbor relation. Previously, it was only known that this mechanism is nearly optimal in the worst case. We then focus on the $\ell_\infty$ neighbor relation, for which the described mechanism is not optimal. We show that one may implement the exponential mechanism for MST in polynomial time, and that this results in universal near-optimality for both the $\ell_1$ and the $\ell_\infty$ neighbor relations.