# The Traveling Salesman Problem 2: 2-opt Removal of Intersections

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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.

Contributed by: Jaime Rangel-Mondragon (July 2012)

Open content licensed under CC BY-NC-SA

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Reference

[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.

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