Private graph colouring with limited 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).
Cite
@article{arxiv.2404.18692,
title = {Private graph colouring with limited defectiveness},
author = {Aleksander B. G. Christiansen and Eva Rotenberg and Teresa Anna Steiner and Juliette Vlieghe},
journal= {arXiv preprint arXiv:2404.18692},
year = {2025}
}