An analogue approach to the travelling salesman problem using an elastic net method

The travelling salesman problem1 is a classical problem in the field of combinatorial optimization, concerned with efficient methods for maximizing or minimizing a function of many independent variables. Given the positions of N cities, which in the simplest case lie in the plane, what is the shortest closed tour in which each city can be visited once? We describe how a parallel analogue algorithm, derived from a formal model2–3 for the establishment of topographically ordered projections in the brain4–10, can be applied to the travelling salesman problem1,11,12. Using an iterative procedure, a circular closed path is gradually elongated non-uniformly until it eventually passes sufficiently near to all the cities to define a tour. This produces shorter tour lengths than another recent parallel analogue algorithm13, scales well with the size of the problem, and is naturally extendable to a large class of optimization problems involving topographic mappings between geometrical structures14.