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Ants find the shortest path: a mathematical proof

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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.

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Notes

  1. 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.

  2. 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.

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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).

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Authors and Affiliations

  1. Department of Electrical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi, 110016, India

    Jayadeva & Suresh Chandra

  2. Thomson Reuters Corp. R & D, 195, Broadway, New York, 10007, USA

    Sameena Shah

  3. Federal University of Rio de Janeiro, Department of Electrical Engineering, PEE/COPPE/UFRJ, P.O. Box 68504, Rio de Janeiro, RJ, 21945-970, Brazil

    Amit Bhaya

  4. IBM India Research Lab., 4 Block C, Institutional Area, Vasant Kunj, New Delhi, 110070, India

    Ravi Kothari

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  1. Jayadeva
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  2. Sameena Shah
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  3. Amit Bhaya
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  4. Ravi Kothari
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  5. Suresh Chandra
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Correspondence to Jayadeva.

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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

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