##### Travelling salesman problem

The **travelling salesman problem** (**TSP**) is an NP-hard problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of cities and their pairwise distances, the task is to find the shortest possible route that visits each city exactly once and returns to the origin city. It is a special case of the travelling purchaser problem.

The problem was first formulated as a mathematical problem in 1930 and is one of the most intensively studied problems in optimization. It is used as a benchmark for many optimization methods. Even though the problem is computationally difficult, a large number of heuristics and exact methods are known, so that some instances with tens of thousands of cities can be solved.

The TSP has several applications even in its purest formulation, such as planning, logistics, and the manufacture of microchips. Slightly modified, it appears as a sub-problem in many areas, such as DNA sequencing. In these applications, the concept *city* represents, for example, customers, soldering points, or DNA fragments, and the concept *distance* represents travelling times or cost, or a similarity measure between DNA fragments. In many applications, additional constraints such as limited resources or time windows make the problem considerably harder.

In the theory of computational complexity, the decision version of the TSP (where, given a length *L*, the task is to decide whether any tour is shorter than *L*) belongs to the class of NP-complete problems. Thus, it is likely that the worst-case running time for any algorithm for the TSP increases exponentially with the number of cities.

The traditional lines of attack for the NP-hard problems are the following:

- Devising algorithms for finding exact solutions (they will work reasonably fast only for small problem sizes).
- Devising “suboptimal” or heuristic algorithms, i.e., algorithms that deliver either seemingly or probably good solutions, but which could not be proved to be optimal.
- Finding special cases for the problem (“subproblems”) for which either better or exact heuristics are possible.

(**Editor’s note:** I love reading this kind of stuff. Researching TSP as it applies to pointillism and printer plotting.)

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