Exploring the k-coloable landscape with Iterated Greedy

Many heuristic algorithms have been proposed for graph coloring. The simplest is perhaps the greedy algorithm. Many variations have been proposed for this algorithm at various levels of sophistication, but it is generally assumed that the coloring will occur in a single attempt. We note that if a new permutation of the vertices is chosen which respects the independent sets of a previous coloring, then applying the greedy algorithm will result in a new coloring in which the number of colors used does not increase, yet may decrease. We introduce several heuristics for generating new permutations that are fast when implemented and eeective in reducing the coloring number. The resulting Iterated Greedy algorithm(IG) can obtain colorings in the range 100 to 103 on graphs in G 1000; 1 2. More interestingly, it can optimally color k-colorable graphs with k up to 60 and n = 1000. We couple this algorithm with several other coloring algorithms, including a modiied TABU search, and one that tries to nd large independent sets using a pruned backtrack. With these combined algorithms we nd 86 and 87 colorings for G 1000; 1 2 , but nd paradoxical results on other classes. Finally, we explore the areas of diiculty in probabilistic graph space under several parameterizations. We check our system on k-colorable graphs in G 300;p;k for 0:05 p 0:95 and 2 k 105. We perform many experiments on G 1000; 1 2 ;k. We nd a narrow ridge where the algorithms fail to nd the speciied coloring, but easy success everywhere else. We test our algorithm on graphs that follow the ridge boundary for p = 1 2 for n up to 10000 vertices.