A tabu search heuristic for the partition coloring problem

Let G = (V, E) be a non-directed graph, where E is the set of edges and V is the set of nodes. Let also V1, V2, . . . , Vp be a partition of V into q subsets such that V1 ∪ V2 ∪ . . . ∪ Vq = V and Vi ∩ Vj = ∅, ∀i, j = 1, . . . , q with i 6= j. We refer to V1, V2, . . . , Vp as the components of the partition. Then, the Partition Coloring Problem (PCP) consists in finding a subset V p ⊂ V such that |V ∩Vi| = 1,∀i = 1, . . . , q (i.e., V p contains one node from each component Vi,∀i = 1, . . . , q), and the chromatic number of the graph induced in G by V p is minimum. This problem is clearly a generalization of the graph coloring problem. Li and Simha [13] have shown that the decision version of PCP is NP-complete.