Abstract
In the most basic application of Ant Colony Optimization (ACO), a set of artificial ants find the shortest path between a source and a destination. Ants deposit pheromone on paths they take, preferring paths that have more pheromone on them. Since shorter paths are traversed faster, more pheromone accumulates on them in a given time, attracting more ants and leading to reinforcement of the pheromone trail on shorter paths. This is a positive feedback process that can also cause trails to persist on longer paths, even when a shorter path becomes available. To counteract this persistence on a longer path, ACO algorithms employ remedial measures, such as using negative feedback in the form of uniform evaporation on all paths. Obtaining high performance in ACO algorithms typically requires fine tuning several parameters that govern pheromone deposition and removal. This paper proposes a new ACO algorithm, called EigenAnt, for finding the shortest path between a source and a destination, based on selective pheromone removal that occurs only on the path that is actually chosen for each trip. We prove that the shortest path is the only stable equilibrium for EigenAnt, which means that it is maintained for arbitrary initial pheromone concentrations on paths, and even when path lengths change with time. The EigenAnt algorithm uses only two parameters and does not require them to be finely tuned. Simulations that illustrate these properties are provided.
Similar content being viewed by others
Notes
This implies that all paths are of different lengths. This assumption can easily be relaxed, at the cost of slightly increased technicalities, without changing the main idea of the proof.
This is because the eigenvalue-eigenvector equation A x=λ x, for a fixed eigenvalue λ has infinitely many solutions. In fact, if x is an eigenvector, then y=γ x, for all γ∈ℝ is also a solution. However, in the present case, there are exactly n solutions, corresponding to n unique values of γ.
References
Abdelbar, A. M., & Wunsch, D. C. (2012). Improving the performance of MAX-MIN ant system on the TSP using stubborn ants. In Proceedings of the fourteenth international conference on genetic and evolutionary computation conference companion, GECCO Companion’12 (pp. 1395–1396). New York: ACM.
Bandieramonte, M., Di Stefano, A., & Morana, G. (2010). Grid jobs scheduling: the alienated ant algorithm solution. Multiagent and Grid Systems, 6(3), 225–243.
Bonabeau, E., Dorigo, M., & Theraulaz, G. (1999). Swarm intelligence: from natural to artificial systems. New York: Oxford University Press.
Chen, L., Sun, H. Y., & Wang, S. (2009). First order deceptive problem of ACO and its performance analysis. Journal of Networks, 4(10), 993–1000.
Deneubourg, J. L., Aron, S., Goss, S., & Pasteels, J. M. (1990). The self-organizing exploratory pattern of the Argentine ant. Journal of Insect Behaviour, 3, 159–168.
Di Caro, G., & Dorigo, M. (1998a). Ant colonies for adaptive routing in packet-switched communications networks. In A. E. Eiben, T. Bäck, M. Schoenauer, & H. P. Schwefel (Eds.), Lecture notes in computer science: Vol. 1498. Parallel problem solving from nature—PPSN V: 5th international conference (pp. 673–682). Berlin: Springer.
Di Caro, G., & Dorigo, M. (1998b). AntNet: distributed stigmergetic control for communications networks. Journal of Artificial Intelligence Research, 9, 317–365.
Ding, Y. Y., He, Y., & Jiang, J. P. (2003). Multi-robot cooperation method based on the ant algorithm. In Proceedings of the swarm intelligence symposium, 2003. SIS’03 (pp. 14–18). New York: IEEE Press.
Dorigo, M. (2007). Ant colony optimization. Scholarpedia, 2(3), 1461. http://www.scholarpedia.org/article/Ant_Colony_Optimization.
Dorigo, M., & Gambardella, L. M. (1997). Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Transactions on Evolutionary Computation, 1(1), 53–66.
Dorigo, M., Maniezzo, V., & Colorni, A. (1996). Ant system: optimization by a colony of cooperating agents. IEEE Transactions Systems, Man, Cybernetics-Part B, 26(1), 29–41.
Dorigo, M., & Stützle, T. (2004). Ant colony optimization. Cambridge: MIT Press.
Ducatelle, F., Di Caro, G. A., & Gambardella, L. M. (2010). Principles and applications of swarm intelligence for adaptive routing in telecommunications networks. Swarm Intelligence, 4(3), 173–198.
Ghazy, A. M., El-Licy, F., & Hefny, H. A. (2012). Threshold based AntNet algorithm for dynamic traffic routing of road networks. Egyptian Informatics Journal, 13(2), 111–121.
Hertz, J., Krogh, A., & Palmer, R. G. (1991). Introduction to the theory of neural computation. Redwood City: Addison-Wesley.
Jackson, D. E., Martin, S. J., Holcombe, M., & Ratnieks, F. L. W. (2006). Longevity and detection of persistent foraging trails in Pharaoh’s ants, Monomorium pharaonis (L.). Animal Behaviour, 71, 351–359.
Jaffe, K. (1980). Theoretical analysis of the communication system for chemical mass recruitment in ants. Journal of Theoretical Biology, 84, 589–609.
Mapisse, J., Cardoso, P., & Monteiro, J. (2011). Ant colony optimization routing mechanisms with bandwidth sensing. In EUROCON—international conference on computer as a tool (pp. 39–42). Lisbon: IEEE Press.
Merkle, D., Middendorf, M., & Schmeck, H. (2002). Ant colony optimization for resource-constrained project scheduling. IEEE Transactions on Evolutionary Computation, 6(4), 333–346.
Meyer, B. (2004). Convergence control in ACO. In Lecture notes in computer science: Vol. 3103. Genetic and evolutionary computation (GECCO) (pp. 1–12), Seattle, WA. Berlin: Springer.
Parpinelli, R. S., Lopes, H. S., & Freitas, A. A. (2002). Data mining with an ant colony optimization algorithm. IEEE Transactions on Evolutionary Computation, 6(4), 321–332.
Reimann, M., Doerner, K., & Hartl, R. F. (2003). Analyzing a unified ant system for the VRP and some of its variants. In Lecture notes in computer science: Vol. 2611. Proceedings of EvoWorkshops: applications of evolutionary computing (pp. 300–310). Berlin: Springer.
Reimann, M., Stummer, M., & Doerner, K. (2002). A savings based ant system for the vehicle routing problem. In Proceedings of the genetic and evolutionary computation conference, GECCO’02 (pp. 1317–1326). San Francisco: Morgan Kaufmann Publishers.
Robinson, E. J. H., Ratnieks, F. L. W., & Holcombe, M. (2008). An agent based model to investigate the roles of attractive and repellent pheromones in ant decision making during foraging. Journal of Theoretical Biology, 255, 250–258.
Shah, S., Kothari, R., & Jayadeva Chandra, S. (2008). Mathematical modeling and convergence analysis of trail formation. In Proceedings of the 23rd national conference on artificial intelligence, advancement of artificial intelligence (AAAI)’08 (Vol. 1, pp. 170–175). Chicago: AAAI Press.
Shah, S., Kothari, R., Jayadeva, & Chandra, S. (2010). Trail formation in ants: a generalized Polya urn process. Swarm Intelligence, 4(2), 145–171.
Stützle, T., & Hoos, H. H. (2000). MAX-MIN ant system. Future Generation Computer Systems, 16(8), 889–914.
Sumpter, D. J. T., & Beekman, M. (2003). From non-linearity to optimality: pheromone trail foraging by ants. Animal Behaviour, 66, 273–280.
Van Vorhis Key, S. E., & Baker, T. C. (1982). Trail-following responses of the Argentine ant Iridomyrmex humilis (Mayr), to a synthetic trail pheromone component and analogs. Journal of Chemical Ecology, 8(1), 3–14.
Yildirim, U. M., & Çatay, B. (2012). A time-based pheromone approach for the ant system. Optimization Letters, 6(6), 1081–1099.
Yuan, Z., Montes de Oca, M. A., Birattari, M., & Stützle, T. (2012). Continuous optimization algorithms for tuning real and integer parameters of swarm intelligence algorithms. Swarm Intelligence, 6(1), 49–75.
Zecchin, A. C., Simpson, A. R., Maier, H. R., & Nixon, J. B. (2005). Parametric study for an ant algorithm applied to water distribution system optimization. IEEE Transactions on Evolutionary Computation, 9(2), 175–191.
Acknowledgements
The authors would like to thank reviewers 2 and 3 for constructive criticism, as well as reviewer 1, the Associate Editors and the Editor for detailed comments and suggestions. The work of the first author (J) was partially supported by a grant from the DST (Indo–Brazil International Collaboration). The work of the last author (AB) was partially supported by grants from FAPERJ (CNE) and CNPq (BPP, Brazil–India International Collaboration).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jayadeva, Shah, S., Bhaya, A. et al. Ants find the shortest path: a mathematical proof. Swarm Intell 7, 43–62 (2013). https://doi.org/10.1007/s11721-013-0076-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11721-013-0076-9