Colouring random graphs and maximising local diversity

TL;DR

This paper investigates a graph colouring variation on random graphs, aiming to maximize local diversity of colours, and adapts algorithms like belief propagation and Walksat to solve it, with practical applications in distributed storage.

Contribution

It introduces a novel graph colouring problem focused on local diversity and adapts existing algorithms to efficiently solve it on random graphs.

Findings

Identifies critical connectivity thresholds for algorithm success

Adapts belief propagation and Walksat for local diversity maximization

Demonstrates practical relevance in distributed storage systems

Abstract

We study a variation of the graph colouring problem on random graphs of finite average connectivity. Given the number of colours, we aim to maximise the number of different colours at neighbouring vertices (i.e. one edge distance) of any vertex. Two efficient algorithms, belief propagation and Walksat are adapted to carry out this task. We present experimental results based on two types of random graphs for different system sizes and identify the critical value of the connectivity for the algorithms to find a perfect solution. The problem and the suggested algorithms have practical relevance since various applications, such as distributed storage, can be mapped onto this problem.