![]() ![]() In the 1950s and 1960s, the problem became increasingly popular in scientific circles in Europe and the United States after the RAND Corporation in Santa Monica offered prizes for steps in solving the problem. The earliest publication using the phrase "travelling salesman problem" was the 1949 RAND Corporation report by Julia Robinson, "On the Hamiltonian game (a traveling salesman problem)." Hassler Whitney at Princeton University generated interest in the problem, which he called the "48 states problem". Flood who was looking to solve a school bus routing problem. It was first considered mathematically in the 1930s by Merrill M. ![]() 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. Rules which would push the number of trials below the number of permutations of the given points, are not known. Of course, this problem is solvable by finitely many trials. We denote by messenger problem (since in practice this question should be solved by each postman, anyway also by many travelers) the task to find, for finitely many points whose pairwise distances are known, the shortest route connecting the points. The general form of the TSP appears to have been first studied by mathematicians during the 1930s in Vienna and at Harvard, notably by Karl Menger, who defines the problem, considers the obvious brute-force algorithm, and observes the non-optimality of the nearest neighbour heuristic: Hamilton's icosian game was a recreational puzzle based on finding a Hamiltonian cycle. The TSP was mathematically formulated in the 19th century by the Irish mathematician William Rowan Hamilton and by the British mathematician Thomas Kirkman. A handbook for travelling salesmen from 1832 mentions the problem and includes example tours through Germany and Switzerland, but contains no mathematical treatment. The origins of the travelling salesman problem are unclear. In many applications, additional constraints such as limited resources or time windows may be imposed. ![]() The TSP also appears in astronomy, as astronomers observing many sources will want to minimize the time spent moving the telescope between the sources in such problems, the TSP can be embedded inside an optimal control problem. 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. Slightly modified, it appears as a sub-problem in many areas, such as DNA sequencing. The TSP has several applications even in its purest formulation, such as planning, logistics, and the manufacture of microchips. Even though the problem is computationally difficult, many 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%. It is used as a benchmark for many optimization methods. The problem was first formulated in 1930 and is one of the most intensively studied problems in optimization. Thus, it is possible that the worst-case running time for any algorithm for the TSP increases superpolynomially (but no more than exponentially) with the number of cities. 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 a tour of at most L) belongs to the class of NP-complete problems. The travelling purchaser problem and the vehicle routing problem are both generalizations of TSP. The travelling salesman problem ( TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in theoretical computer science and operations research. Solution of a travelling salesman problem: the black line shows the shortest possible loop that connects every red dot. NP-hard problem in combinatorial optimization
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