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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 ...
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 ...
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.
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 ...
For example, the optimization problem of finding the least-cost cyclic route through all nodes of a weighted graph—commonly known as the travelling salesman problem—is NP-hard. [7] The subset sum problem is another example: given a set of integers, does
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 ...
The algorithm builds a tour for the traveling salesman one edge at a time and thus maintains multiple tour fragments, each of which is a simple path in the complete graph of cities. At each stage, the algorithm selects the edge of minimal cost that either creates a new fragment, extends one of the existing paths or creates a cycle of length ...
The travelling salesman problem asks to find the shortest cyclic tour of a collection of points, in the plane or in more abstract mathematical spaces. Because the problem is NP-hard, algorithms that take polynomial time are unlikely to be guaranteed to find its optimal solution; [2] on the other hand a brute-force search of all permutations would always solve the problem exactly but would take ...