Approximation Algorithms for the Traveling Salesman Problem with Range Condition

D. Arun Kumar; C. Pandu Rangan

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 34, Issue: 3, page 173-181
  • ISSN: 0988-3754

Abstract

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We prove that the Christofides algorithm gives a 4 3 approximation ratio for the special case of traveling salesman problem (TSP) in which the maximum weight in the given graph is at most twice the minimum weight for the odd degree restricted graphs. A graph is odd degree restricted if the number of odd degree vertices in any minimum spanning tree of the given graph is less than 1 4 times the number of vertices in the graph. We prove that the Christofides algorithm is more efficient (in terms of runtime) than the previous existing algorithms for this special case of the traveling salesman problem. Secondly, we apply the concept of stability of approximation to this special case of traveling salesman problem in order to partition the set of all instances of TSP into an infinite spectrum of classes according to their approximability.

How to cite

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D. Arun Kumar, and Pandu Rangan, C.. " Approximation Algorithms for the Traveling Salesman Problem with Range Condition." RAIRO - Theoretical Informatics and Applications 34.3 (2010): 173-181. <http://eudml.org/doc/221984>.

@article{D2010,
abstract = { We prove that the Christofides algorithm gives a $\frac\{4\}\{3\}$ approximation ratio for the special case of traveling salesman problem (TSP) in which the maximum weight in the given graph is at most twice the minimum weight for the odd degree restricted graphs. A graph is odd degree restricted if the number of odd degree vertices in any minimum spanning tree of the given graph is less than $\frac\{1\}\{4\}$ times the number of vertices in the graph. We prove that the Christofides algorithm is more efficient (in terms of runtime) than the previous existing algorithms for this special case of the traveling salesman problem. Secondly, we apply the concept of stability of approximation to this special case of traveling salesman problem in order to partition the set of all instances of TSP into an infinite spectrum of classes according to their approximability. },
author = {D. Arun Kumar, Pandu Rangan, C.},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {odd degree restricted graphs},
language = {eng},
month = {3},
number = {3},
pages = {173-181},
publisher = {EDP Sciences},
title = { Approximation Algorithms for the Traveling Salesman Problem with Range Condition},
url = {http://eudml.org/doc/221984},
volume = {34},
year = {2010},
}

TY - JOUR
AU - D. Arun Kumar
AU - Pandu Rangan, C.
TI - Approximation Algorithms for the Traveling Salesman Problem with Range Condition
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 3
SP - 173
EP - 181
AB - We prove that the Christofides algorithm gives a $\frac{4}{3}$ approximation ratio for the special case of traveling salesman problem (TSP) in which the maximum weight in the given graph is at most twice the minimum weight for the odd degree restricted graphs. A graph is odd degree restricted if the number of odd degree vertices in any minimum spanning tree of the given graph is less than $\frac{1}{4}$ times the number of vertices in the graph. We prove that the Christofides algorithm is more efficient (in terms of runtime) than the previous existing algorithms for this special case of the traveling salesman problem. Secondly, we apply the concept of stability of approximation to this special case of traveling salesman problem in order to partition the set of all instances of TSP into an infinite spectrum of classes according to their approximability.
LA - eng
KW - odd degree restricted graphs
UR - http://eudml.org/doc/221984
ER -

References

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  8. H.N. Gabow and R.E. Tarjan, Faster scaling algorithms for general graph-matching problems. J. ACM28 (1991) 815-853.  Zbl0799.68145
  9. J. Hromkovic, Stability of approximation algorithms for hard optimisation problems, in Proc. SOFSEM'99. Springer-Verlag, Lecture Notes in Comput. Sci.1725 (1999) 29-46.  Zbl0961.65060
  10. J. Hromkovic, Stability of approximation algorithms and the knapsack problem, in Jewels are forever, edited by J. Karhumäki, H. Maurer and G. Rozenberg. Springer-Verlag (1999) 238-249.  Zbl0945.68074
  11. C.H. Papadimitriou, Euclidean TSP is NP-complete. TCS4 (1977) 237-244.  Zbl0386.90057
  12. C.H. Papadimitriou and M. Yannakakis, The Traveling salesman problem with distances one and two. Math. Oper. Res.18 (1993) 1-11.  Zbl0778.90057

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