A graph coloring heuristic using partial solutions and a reactive tabu scheme

Abstract

Most of the recent heuristics for the graph coloring problem start from an infeasible k-coloring (adjacent vertices may have the same color) and try to make the solution feasible through a sequence of color exchanges. In contrast, our approach (called F OO-P ARTIALCOL), which is based on tabu search, considers feasible but partial solutions and tries to increase the size of the current partial solution. A solution consists of k disjoint stable sets (and, therefore, is a feasible, partial k-coloring) and a set of uncolored vertices. We introduce a reactive tabu tenure which substantially enhances the performance of both our heuristic as well as the classical tabu algorithm (called T ABUCOL) proposed by Hertz and de Werra [Using tabu search techniques for graph coloring, Computing 1987;39:345–51]. We will report numerical results on different benchmark graphs and we will observe that F OO-P ARTIALCOL, though very simple, outperforms T ABUCOL on some instances, provides very competitive results on a set of benchmark graphs which are known to be difficult, and outperforms the best-known methods on the graph flat300_28_0. For this graph, F OO-P ARTIALCOL finds an optimal coloring with 28 colors. The best coloring achieved to date uses 31 colors. Algorithms very close to T ABUCOL are still used as intensification procedures in the best coloring methods, which are evolutionary heuristics. F OO-P ARTIALCOL could then be a powerful alternative. In conclusion F OO-P ARTIALCOL is one of the most efficient simple local search coloring methods yet available.