Near-Universally-Optimal Differentially Private Minimum Spanning Trees

TL;DR

This paper introduces a differentially private algorithm for approximating minimum spanning trees that is nearly optimal in a universal, input-dependent sense, improving upon worst-case guarantees for graph privacy.

Contribution

It presents the first universal optimality result for differentially private MST algorithms, including polynomial-time implementations for both  and inity neighbor relations.

Findings

The simple mechanism is near-optimal for  neighbor relation.

Exponential mechanism achieves universal near-optimality for both  and inity.

The approach advances input-dependent privacy guarantees in graph algorithms.

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.