Travelling salesman problem ant system algorithm pheromone updating
The travelling purchaser problem and the vehicle routing problem are both generalizations of TSP.
In the theory of computational complexity, the decision version of the TSP (where, given a length L, the task is to decide whether the graph has any tour shorter than L) belongs to the class of NP-complete problems.
They found they only needed 26 cuts to come to a solution for their 49 city problem.
While this paper did not give an algorithmic approach to TSP problems, the ideas that lay within it were indispensable to later creating exact solution methods for the TSP, though it would take 15 years to find an algorithmic approach in creating these cuts.
Slightly modified, it appears as a sub-problem in many areas, such as DNA sequencing.
The rule that one first should go from the starting point to the closest point, then to the point closest to this, etc., in general does not yield the shortest route.
Notable contributions were made by George Dantzig, Delbert Ray Fulkerson and Selmer M.
It is used as a benchmark for many optimization methods.
Even though the problem is computationally difficult, a large number of heuristics and exact algorithms are known, so that some instances with tens of thousands of cities can be solved completely and even problems with millions of cities can be approximated within a small fraction of 1%.