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Cellular Learning Automata-based Graph Coloring Problem

The vertex coloring problem is a well-known classical problem in graph theory in which a color is assigned to each vertex of the graph such that no two adjacent vertices have the same color. The minimum vertex coloring problem is known to be an NP-hard problem in an arbitrary graph. In this paper, an irregular cellular learning automata (ICLA) based approximation algorithm is proposed for solving the minimum (vertex) coloring problem. It is shown that by a proper choice of the parameters of the algorithm, the probability of approximating the optimal solution is as close to unity as possible. Our proposed algorithm is compared with some well-known coloring algorithms and the results show the superiority of the proposed algorithm both in terms of the color set size and running time of algorithm over the existing methods. A given graph G is p-colorable, if it can be legally colored with at most p different colors. The chromatic number χ(G) is the minimum number of colors required for coloring the graph, and a graph G is said to be p- chromatic, if χ(G)=p. A minimum coloring of G is a legal coloring in which the smallest number of colors (i.e., chromatic number) to be assigned to the vertices. In (7), Akbari Torkestani and Meybodi proposed four approximation algorithms for solving the graph coloring problem based on learning automata, when each automaton updates its action probability vector using a LR-I reinforcement scheme. In this paper, we propose an algorithm based on irregular cellular learning automata for solving this problem. The proposed algorithm is compared with the algorithms of Caramia(8), Funabiki(9) and Malaguti(10). The Computation results indicate that the proposed algorithm provides better results obtained from color set size, but in running time, the above mentioned algorithms has a better performance.