Travelling salesman problem ant system algorithm pheromone updating
In the following decades, the problem was studied by many researchers from mathematics, computer science, chemistry, physics, and other sciences.
In the 1960s however a new approach was created, that instead of seeking optimal solutions, one would produce a solution whose length is provably bounded by a multiple of the optimal length, and in doing so create lower bounds for the problem; these may then be used with branch and bound approaches.
They found they only needed 26 cuts to come to a solution for their 49 city problem.
Gerhard Reinelt published the TSPLIB in 1991, a collection of benchmark instances of varying difficulty, which has been used by many research groups for comparing results.
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.
In many applications, additional constraints such as limited resources or time windows may be imposed.
The origins of the travelling salesman problem are unclear.