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travelling_salesman_problem

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travelling_salesman_problem [2017/03/27 05:59] ntcnet [Computational complexity] |
travelling_salesman_problem [2018/10/01 13:29] (current) bsuresh old revision restored (2017/04/05 23:01) |
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===== Computing a solution ===== | ===== Computing a solution ===== | ||

- | <note tip>tip</note>==== Computational complexity ==== | + | ==== Computational complexity ==== |

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 ("given the costs and a number x, decide whether there is a round-trip route cheaper than x") is NP-complete. | 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 ("given the costs and a number x, decide whether there is a round-trip route cheaper than x") is NP-complete. | ||

=== 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 [[http://www.bantayso.com|website]], 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, the problem becomes APX-complete and Christofides’s algorithm approximates it within 1.5. |

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. |

travelling_salesman_problem.1490608751.txt.gz · Last modified: 2017/03/27 05:59 by ntcnet

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