An Analysis of a Simple Local Search Algorithm for Graph Colouring

TL;DR

This paper provides an analytical study of the Vertex Descent local search algorithm for graph colouring, showing it can find feasible solutions efficiently for certain graph classes but may fail in others.

Contribution

It offers the first theoretical analysis of Vertex Descent, demonstrating polynomial-time success on specific graph types and highlighting its limitations.

Findings

Vertex Descent finds feasible colourings in expected polynomial time for paths and certain degree-3 graphs.

It can solve 3-colouring problems efficiently for some graphs where heuristics like DSATUR need more colours.

Vertex Descent may fail with high probability on forests with maximum degree 3.

Abstract

Vertex Descent is a local search algorithm which forms the basis of a wide spectrum of tabu search, simulated annealing and hybrid evolutionary algorithms for graph colouring. These algorithms are usually treated as experimental and provide strong results on established benchmarks. As a step towards studying these heuristics analytically, an analysis of the behaviour of Vertex Descent is provided. It is shown that Vertex Descent is able to find feasible colourings for several types of instances in expected polynomial time. This includes 2-colouring of paths and 3-colouring of graphs with maximum degree 3. The same also holds for 3-colouring of a subset of 3-colourable graphs with maximum degree 4. As a consequence, Vertex Descent finds a 3-colouring in expected polynomial time for the smallest graph for which Br\'elaz's heuristic DSATUR needs 4 colours. On the other hand, Vertex Descent may fail for forests with maximum degree 3 with high probability.