The travelling salesman problem (TSP) is an NP-hard

NP-hard

NP-hard , in computational complexity theory, is a class of problems that are, informally, "at least as hard as the hardest problems in NP". A problem H is NP-hard if and only if there is an NP-complete problem L that is polynomial time Turing-reducible to H...

 problem in combinatorial optimization

Combinatorial optimization

In applied mathematics and theoretical computer science, combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects. In many such problems, exhaustive search is not feasible...

 studied in operations research

Operations research

Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...

 and theoretical computer science

Theoretical computer science

Theoretical computer science is a division or subset of general computer science and mathematics which focuses on more abstract or mathematical aspects of computing....

. Given a list of cities and their pairwise distances, the task is to find the shortest possible tour that visits each city exactly once. It is a special case of the Traveling purchaser problem

Traveling purchaser problem

The traveling purchaser problem is an NP-hard problem studied in theoretical computer science. Given a list of market-places, their distances and the cost of the available articles, the task is to find a route which minimizes the cost for buying a certain set of articles available at differing...

.

The problem was first formulated as a mathematical problem in 1930 and is one of the most intensively studied problems in optimization.