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travelling_salesman_problem [2014/09/23 22:48] sertalpbilal |
travelling_salesman_problem [2024/04/29 14:19] 47.128.53.165 old revision restored (2024/04/26 14:15) |
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====== Travelling salesman problem ====== | ====== Travelling salesman problem ====== | ||
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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, | 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, | ||
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===== Computing a solution ===== | ===== Computing a solution ===== | ||
- | ==== Computational complexity ==== | + | <note tip> |
The problem has been shown to be NP-hard (more precisely, it is complete for the complexity class FPNP; see function problem), and the decision problem version (" | The problem has been shown to be NP-hard (more precisely, it is complete for the complexity class FPNP; see function problem), and the decision problem version (" | ||
=== Complexity of approximation === | === Complexity of approximation === | ||
- | In the general case, finding a shortest travelling salesman tour is NPO-complete. If the distance measure is a metric and symmetric, the problem becomes APX-complete and Christofides’s algorithm approximates it within 1.5. | + | In the general case, finding a shortest travelling salesman tour is NPO-complete. If the distance measure is a metric and symmetric |
If the distances are restricted to 1 and 2 (but still are a metric) the approximation ratio becomes 8/7. In the asymmetric, metric case, only logarithmic performance guarantees are known, the best current algorithm achieves performance ratio $0.814 log(n)$; it is an open question if a constant factor approximation exists. | If the distances are restricted to 1 and 2 (but still are a metric) the approximation ratio becomes 8/7. In the asymmetric, metric case, only logarithmic performance guarantees are known, the best current algorithm achieves performance ratio $0.814 log(n)$; it is an open question if a constant factor approximation exists. |