The Traveling Salesman Problem 2: 2-opt Removal of Intersections
The traveling salesman problem (TSP) is the most famous combinatorial optimization problem. Its task is to find a tour through a set of vertices in the plane with the shortest possible total length. This Demonstration explores a method that helps in attaining the optimal minimum. It is based on the removal of intersections in a path to decrease its length, relying on the following fact: given points on the plane, no three of which are collinear, there exists a closed path with no self-intersections having those points as vertices of minimal length. This method is called 2-opt and was proposed in 1958 as a way to solve the TSP. Although 2-opt does not give the optimum path, it often improves a given path. This Demonstration generates a random path and applies the method repeatedly to it, comparing the initial and final lengths obtained.
 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.