On the Invariance of Ant Colony Optimization for the Traveling Salesman Problem

Ant colony optimization (ACO) is nowadays one of the most promising metaheuristics, and an increasing amount of research has been devoted to its empirical and theoretical analysis. Some authors believe that the performance of ant colony optimization depends somehow on the scale of the problem instance under analysis. The issue has been recently raised explicitly [1] and the hyper-cube framework has been proposed to handle this supposed dependency of ACO on the scale of the instances. This paper shows that the ACO internal state—commonly referred to as the pheromone in the literature—indeed depends on the scale of the problem at hand. Nonetheless, we formally prove that this does not affect the external behavior of the algorithm. In other words, the sequence of solutions produced by ACO does not depend on the scale of the problem instance under analysis. Moreover, the paper introduces three variations of the three most widely adopted algorithms belonging to the ant colony optimization family. We formally show that the algorithms we propose are functionally equivalent to the original ones, that is, for any given instance, these algorithms produce the same sequence of solutions as the original ones. Nonetheless, in these new algorithms, also the internal state is independent of the scale of the problem instance at hand.