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Solution of a travelling salesman problem: the black line shows the shortest possible loop that connects every red dot. In the theory of computational complexity, 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 ...
www.math.uwaterloo.ca /tsp /concorde.html The Concorde TSP Solver is a program for solving the travelling salesman problem . It was written by David Applegate , Robert E. Bixby , Vašek Chvátal , and William J. Cook , in ANSI C , and is freely available for academic use.
There exist inputs to the travelling salesman problem that cause the Christofides algorithm to find a solution whose approximation ratio is arbitrarily close to 3/2. One such class of inputs are formed by a path of n vertices, with the path edges having weight 1 , together with a set of edges connecting vertices two steps apart in the path with ...
Greedy algorithms fail to produce the optimal solution for many other problems and may even produce the unique worst possible solution. One example is the travelling salesman problem mentioned above: for each number of cities, there is an assignment of distances between the cities for which the nearest-neighbour heuristic produces the unique ...
A function problem is a computational problem where a single output (of a total function) is expected for every input, but the output is more complex than that of a decision problem—that is, the output is not just yes or no. Notable examples include the traveling salesman problem and the integer factorization problem.
The Held–Karp algorithm, also called the Bellman–Held–Karp algorithm, is a dynamic programming algorithm proposed in 1962 independently by Bellman [1] and by Held and Karp [2] to solve the traveling salesman problem (TSP), in which the input is a distance matrix between a set of cities, and the goal is to find a minimum-length tour that visits each city exactly once before returning to ...
In the worst case, the algorithm results in a tour that is much longer than the optimal tour. To be precise, for every constant r there is an instance of the traveling salesman problem such that the length of the tour computed by the nearest neighbour algorithm is greater than r times the length of the optimal tour. Moreover, for each number of ...
In combinatorial optimization, Lin–Kernighan is one of the best heuristics for solving the symmetric travelling salesman problem. [citation needed] It belongs to the class of local search algorithms, which take a tour (Hamiltonian cycle) as part of the input and attempt to improve it by searching in the neighbourhood of the given tour for one that is shorter, and upon finding one repeats the ...