An Analysis of the Elastic Net Approach to the Travelling Salesman Problem
This paper analyzes the elastic net approach (Durbin and Willshaw 1987) to the traveling salesman problem of finding the shortest path through a set of cities. The elastic net approach jointly minimizes the length of an arbitrary path in the plane and the distance between the path points and the cities. The tradeoff between these two requirements is controlled by a scale parameter K. A global minimum is found for large K, and is then tracked to a small value. In this paper, we show that (1) in the small K limit the elastic path passes arbitrarily close to all the cities, but that only one path point is attracted to each city, (2) in the large K limit the net lies at the center of the set of cities, and (3) at a critical value of K the energy function bifurcates. We also show that this method can be interpreted in terms of extremizing a probability distribution controlled by K. The minimum at a given K corresponds to the maximum a posteriori (MAP) Bayesian estimate of the tour under a natural statistical interpretation. The analysis presented in this paper gives us a better understanding of the behavior of the elastic net, allows us to better choose the parameters for the optimization, and suggests how to extend the underlying ideas to other domains.