Private graph colouring with limited defectiveness

TL;DR

This paper investigates the limits and possibilities of differentially private vertex colouring in graphs, establishing lower bounds on defectiveness and presenting an algorithm that balances privacy, colour count, and defectiveness.

Contribution

It provides the first lower bound on defectiveness for private graph colouring and introduces an algorithm achieving near-optimal defectiveness under differential privacy constraints.

Findings

Lower bound on defectiveness: d = Ω(log n / (log c + log Δ)).

An ε-differentially private algorithm with defectiveness Θ(log n).

Trade-off between privacy, number of colours, and defectiveness.

Abstract

Differential privacy is the gold standard in the problem of privacy preserving data analysis, which is crucial in a wide range of disciplines. Vertex colouring is one of the most fundamental questions about a graph. In this paper, we study the vertex colouring problem in the differentially private setting. To be edge-differentially private, a colouring algorithm needs to be defective: a colouring is d-defective if a vertex can share a colour with at most d of its neighbours. Without defectiveness, the only differentially private colouring algorithm needs to assign n different colours to the n different vertices. We show the following lower bound for the defectiveness: a differentially private c-edge colouring algorithm of a graph of maximum degree {\Delta} > 0 has defectiveness at least d = {\Omega} (log n / (log c+log {\Delta})). We also present an {\epsilon}-differentially private algorithm to {\Theta} ( {\Delta} / log n + 1 / {\epsilon})-colour a graph with defectiveness at most {\Theta}(log n).