An Ant Algorithm for the Graph Colouring Problem

This paper describes an ant algorithm that colours graphs in an e cient way. First, we describe the graph colouring problem and the ant algorithm. We also present some results and propose a simple generalisation of the algorithm that might allow its application to other assignment problems. Finally, a short study of the algorithm operators explains its performance and shows that it may be seen as a parallel variation of tabu search, with an implicit memory instead of an avoidance list. ANTS'98 From Ant Colonies to Arti cial Ants: First International Workshop on Ant Colony Optimization Brussels, Belgium, October 15{16, 1998. c F.Comellas and J. Oz on. 1 An ant algorithm for colouring graphs Many assignment and combinatorial problems may be formulated in terms of graph colouring. A proper colouring of a graph G = (V;E) is a function from the nodes of the graph to a set C of colours such that any two adjacent nodes have di erent colours. If k is the cardinal of C, we say that G is k-coloured. The graph colouring problem is NP-complete. Hence, we need to use approximate algorithmic methods to obtain solutions close to the absolute minimum in a reasonable execution time. Most of the algorithms used to solve the colouring problem are summarised in Hurley [4], and include sequential assignment methods, simulated annealing and genetic algorithms. The ant algorithm that we propose for the colouring problem is a multi-agent system based on the idea of parallel search. In this algorithm a given number of ants move around the nodes of the graph and changes the colour of each visited node according to a local criterion. At a given iteration each ant moves from the current node to the adjacent node with the maximum number of violations (see Fig 1), and replaces the old colour of the node with a new colour that minimises this number. For a given node i, the number of violations is computed as the number of adjacent nodes to i with the same colour than i. This action is randomly repeated for each ant: the ant moves to the