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**Abstract**: Standard *Stigmergic* approaches to *Swarm Intelligence* encompasses the use of a set of stochastic cooperating ant-like agents to find optimal solutions, using self-organized * Stigmergy* as an indirect form of communication mediated by a singular artificial pheromone. Agents deposit pheromone-signs on the edges of the problem-related graph to give rise to a family of successful algorithmic approaches entitled

*Ant Systems*(

**AS**),

*Ant Colony Systems*(

**ACS**), among others. These mainly rely on positive feedbacks, to search for an optimal solution in a large combinatorial space. The present work shows how, using two different sets of pheromones, a second-order coevolved compromise between positive and negative feedbacks achieves better results that single positive feedback systems. This follows the route of very recent biological findings showing that forager ants, while laying attractive trail pheromones to guide nest mates to food, also gained foraging effectiveness by the use of pheromones that repelled foragers from unrewarding routes. The algorithm presented here takes inspiration from this biological observation.

The new algorithm was exhaustively tested on a series of well-known benchmarks over hard NP-complete *Combinatorial Optimization Problems* (**COP**’s), running on symmetrical *Traveling Salesman Problems* (**TSP**). Different network topologies and stress tests were conducted over low-size TSP’s (eil51.tsp; eil78.tsp; kroA100.tsp), medium-size (d198.tsp; lin318.tsp; pcb442.tsp; att532.tsp; rat783.tsp) as well as large sized ones (fl1577.tsp; d2103.tsp) [numbers here referring to the number of nodes in the network]. We show that the new co-evolved stigmergic algorithm compared favorably against the benchmark. The algorithm was able to equal or majorly improve every instance of those standard algorithms, not only in the realm of the Swarm Intelligent **AS**, **ACS** approach, as in other computational paradigms like *Genetic Algorithms* (**GA**), *Evolutionary Programming* (**EP**), as well as **SOM** (*Self-Organizing Maps*) and **SA** (*Simulated Annealing*). In order to deeply understand how a second co-evolved pheromone was useful to track the collective system into such results, a refined phase-space map was produced mapping the pheromones ratio between a pure *Ant Colony System* (where no negative feedback besides pheromone evaporation is present) and the present second-order approach. The evaporation rate between different pheromones was also studied and its influence in the outcomes of the algorithm is shown. A final discussion on the phase-map is included. This work has implications in the way large combinatorial problems are addressed as the double feedback mechanism shows improvements over the single-positive feedback mechanisms in terms of convergence speed and of major results.

**Keywords**: Stigmergy, Co-Evolution, Self-Organization, Swarm Intelligence, Foraging, Cooperative Learning, Combinatorial Optimization problems, Symmetrical Traveling Salesman Problems (TSP), phase-space.