A hash function breaking symmetry in partitioning problems and its application to tabu search for graph coloring

In partitioning problems we are asked to divide a set of elements into groups such that some properties are satisfied. Local search algorithms solve these problems representing a candidate solution by an array that maps each element to one of the groups. Yet the name of the group is not a relevant information and a simple permutation of the labels of the groups yields an equivalent solution. In this note we introduce a hash function that takes into account this symmetry of the search space and can be used to prevent revisiting solutions that differ only for a permutation of group labels. We use the function to store and recognize solutions in a tabu search algorithm for graph coloring and we compare it experimentally with a more common version of this algorithm that simply prevents the reintroduction of a single attribute.