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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.
The cost of the solution produced by the algorithm is within 3/2 of the optimum. To prove this, let C be the optimal traveling salesman tour. Removing an edge from C produces a spanning tree, which must have weight at least that of the minimum spanning tree, implying that w(T) ≤ w(C) - lower bound to the cost of the optimal solution.
The objective of a heuristic is to produce a solution in a reasonable time frame that is good enough for solving the problem at hand. This solution may not be the best of all the solutions to this problem, or it may simply approximate the exact solution. But it is still valuable because finding it does not require a prohibitively long time.
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 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 ...
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 ...
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 ...