The Traveling Salesman Problem 3: Nearest Neighbor Heuristic

Because of its simplicity, the nearest neighbor heuristic is one of the first algorithms that comes to mind in attempting to solve the traveling salesman problem (TSP), in which a salesman has to plan a tour of cities that is of minimal length. In this heuristic, the salesman starts at some city and then visits the city nearest to the starting city, and so on, only taking care not to visit a city twice. At the end all cities are visited and the salesman returns to the starting city.

This Demonstration displays nearest neighbor tours (use the step slider to see them) along with a better tour computed by the built-in Mathematica function FindShortestTour, forming the outline of the blue polygon. All nearest neighbor tours start at point 1.

There is a moment at which some points are "forgotten" during the course of the algorithm, and they have to be inserted at a great cost in the end. Though usually rather bad, nearest neighbor tours have the advantage that they only contain a few severe mistakes while being very fast and easy to implement. Therefore, such tours can serve as good starting tours that other methods can improve.

[1] E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys, eds., The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, New York: John Wiley & Sons, 1985.