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Approximation algorithms for the traveling salesman problem

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Date
2003
Dewey
Recherche opérationnelle
Sujet
Approximation algorithms; NP-complete problem; Traveling salesman
Journal issue
Mathematical Methods of Operations Research
Volume
56
Number
3
Publication date
01-2003
Article pages
387-405
Publisher
Springer
DOI
http://dx.doi.org/10.1007/s001860200239
URI
https://basepub.dauphine.fr/handle/123456789/2980
Collections
  • LAMSADE : Publications
Metadata
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Author
Paschos, Vangelis
Monnot, Jérôme
Toulouse, Sophie
Type
Article accepté pour publication ou publié
Abstract (EN)
We first prove that the minimum and maximum traveling salesman problems, their metric versions as well as some versions defined on parameterized triangle inequalities (called sharpened and relaxed metric traveling salesman) are all equi-approximable under an approximation measure, called differential-approximation ratio, that measures how the value of an approximate solution is placed in the interval between the worst- and the best-value solutions of an instance. We next show that the 2OPT, one of the most-known traveling salesman algorithms, approximately solves all these problems within differential-approximation ratio bounded above by 1/2. We analyze the approximation behavior of 2OPT when used to approximately solve traveling salesman problem in bipartite graphs and prove that it achieves differential-approximation ratio bounded above by 1/2 also in this case. We also prove that, for any ε>0, it is NP-hard to differentially approximate metric traveling salesman within better than 649/650 + ε and traveling salesman with distances 1 and 2 within better than 741/742 + ε. Finally, we study the standard approximation of the maximum sharpened and relaxed metric traveling salesman problems. These are versions of maximum metric traveling salesman defined on parameterized triangle inequalities and, to our knowledge, they have not been studied until now.

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