1 ACO Algorithms for the Traveling Salesman Problem

Ant algorithms [18, 14, 19] are a recently developed, population-based approach which has been successfully applied to several NP-hard combinatorial optimization problems [6, 13, 17, 23, 34, 40, 49]. As the name suggests, ant algorithms have been inspired by the behavior of real ant colonies, in particular, by their foraging behavior. One of the main ideas of ant algorithms is the indirect communication of a colony of agents, called (artificial) ants, based on pheromone trails (pheromones are also used by real ants for communication). The (artificial) pheromone trails are a kind of distributed numeric information which is modified by the ants to reflect their experience while solving a particular problem. Recently, the Ant Colony Optimization (ACO) metaheuristic has been proposed which provides a unifying framework for most applications of ant algorithms [15, 16] to combinatorial optimization problems. In particular, all the ant algorithms applied to the TSP fit perfectly into the ACO meta-heuristic and, therefore, we will call these algorithms also ACO algorithms. The first ACO algorithm, called Ant System (AS) [18, 14, 19], has been applied to the Traveling Salesman Problem (TSP). Starting from Ant System, several improvements of the basic algorithm have been proposed [21, 22, 17, 51, 53, 7]. Typically, these improved algorithms have been tested again on the TSP. All these improved versions of AS have in common a stronger exploita-