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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