Palette-colouring: a belief-propagation approach

TL;DR

This paper develops an advanced belief propagation algorithm for the palette-colouring problem, a graph colouring variation, analyzing its phase transition behavior on large sparse random graphs.

Contribution

It introduces a correct belief propagation method considering many-body constraints, improving upon naive approaches for palette-colouring.

Findings

The belief propagation algorithm effectively solves palette-colouring under certain conditions.

A phase transition from satisfiable to unsatisfiable states occurs as vertex degree increases.

The method's performance depends on graph ensemble and size, revealing critical thresholds.

Abstract

We consider a variation of the prototype combinatorial-optimisation problem known as graph-colouring. Our optimisation goal is to colour the vertices of a graph with a fixed number of colours, in a way to maximise the number of different colours present in the set of nearest neighbours of each given vertex. This problem, which we pictorially call "palette-colouring", has been recently addressed as a basic example of problem arising in the context of distributed data storage. Even though it has not been proved to be NP complete, random search algorithms find the problem hard to solve. Heuristics based on a naive belief propagation algorithm are observed to work quite well in certain conditions. In this paper, we build upon the mentioned result, working out the correct belief propagation algorithm, which needs to take into account the many-body nature of the constraints present in this problem. This method improves the naive belief propagation approach, at the cost of increased computational effort. We also investigate the emergence of a satisfiable to unsatisfiable "phase transition" as a function of the vertex mean degree, for different ensembles of sparse random graphs in the large size ("thermodynamic") limit.