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It is easy to prove that the last step works. Moreover, thanks to the triangle inequality, each skipping at Step 4 is in fact a shortcut, i.e., the length of the cycle does not increase. Hence it gives us a TSP tour no more than twice as long as the optimal one.

The length of the of the network is a natural lower bound for the length of the optimal route. In the TSP with case it is possible to prove upper bounds in terms of the and design an algorithm that has a provable upper bound on the length of the route. The first published (and the simplest) example follows.

An exact solution for 15,112 German towns from TSPLIB was found in 2001 using the proposed by , , and Selmer Johnson in 1954, based on . The computations were performed on a network of 110 processors located at and (see the Princeton external link). The total computation time was equivalent to 22.6 years on a single 500 MHz . In May 2004, the travelling salesman problem of visiting all 24,978 towns in Sweden was solved: a tour of length approximately 72,500 kilometers was found and it was proven that no shorter tour exists.^{}

In March 2005, the travelling salesman problem of visiting all 33,810 points in a circuit board was solved using : a tour of length 66,048,945 units was found and it was proven that no shorter tour exists. The computation took approximately 15.7 CPU years (Cook et al. 2006). In April 2006 an instance with 85,900 points was solved using , taking over 136 CPU years, see .

Improving these time bounds seems to be difficult. For example, it is an open problem if there exists an exact algorithm for TSP that runs in time ^{}

The presented paper research works have been devoted to ant colony optimization problems such as traveling salesman problem (TSP).

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research works have been devoted to ant colony optimization if a salesman starting.(TSP) we know of was made in 1930 by the Viennese mathematician Karl other portions, of this paper may be copied or distributed royalty ftee without further that most of the research on optimization methods done at.

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Laporte / The traveling salesman problem: Overview of algorithms.A Survey Paper on Solving Travelling Salesman problem Using Bee colony optimization to Traveling Salesman Problem (TSP).

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The **Travelling Salesman problem** (**TSP**) is a problem in studied in and . Given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each city exactly once.

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