On the Invariance of Ant System

It is often believed that the performance of ant system, and in general of ant colony optimization algorithms, depends somehow on the scale of the problem instance at hand. The issue has been recently raised explicitly [1] and the hyper-cube framework has been proposed to eliminate this supposed dependency. In this paper, we show that although the internal state of ant system— that is, the pheromone matrix—depends on the scale of the problem instance under analysis, this does not affect the external behavior of the algorithm. In other words, for an appropriate initialization of the pheromone, the sequence of solutions obtained by ant system does not depend on the scale of the instance. As a second contribution, the paper introduces a straightforward variant of ant system in which also the pheromone matrix is independent of the scale of the problem instance under analysis.