Tighter Bounds for Local Differentially Private Core Decomposition and Densest Subgraph

TL;DR

This paper advances the understanding of differentially private core decomposition by establishing lower bounds and providing mechanisms with improved accuracy, also enhancing solutions for the densest subgraph problem in the local privacy setting.

Contribution

It introduces the first lower bounds on additive error for private core decomposition and presents mechanisms with nearly optimal accuracy in the local differential privacy model.

Findings

Established lower bounds on additive error for private core decomposition.

Developed mechanisms with nearly matching error bounds for exact and approximate core decomposition.

Improved mechanisms for the approximate densest subgraph problem in the local privacy setting.

Abstract

Computing the core decomposition of a graph is a fundamental problem that has recently been studied in the differentially private setting, motivated by practical applications in data mining. In particular, Dhulipala et al. [FOCS 2022] gave the first mechanism for approximate core decomposition in the challenging and practically relevant setting of local differential privacy. One of the main open problems left by their work is whether the accuracy, i.e., the approximation ratio and additive error, of their mechanism can be improved. We show the first lower bounds on the additive error of approximate and exact core decomposition mechanisms in the centralized and local model of differential privacy, respectively. We also give mechanisms for exact and approximate core decomposition in the local model, with almost matching additive error bounds. Our mechanisms are based on a black-box application of continual counting. They also yield improved mechanisms for the approximate densest subgraph problem in the local model.