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Approximation Algorithms for the Traveling Salesman Problem with Range Condition

Published online by Cambridge University Press:  15 April 2002

D. Arun Kumar and
C. Pandu Rangan
D. Arun Kumar
Affiliation:
Lehrstuhl für Informatik I, RWTH Aachen, 52056 Aachen, GermanyDepartment of Computer-Science, Indian Institute of Technology, Madras 600036, India
C. Pandu Rangan
Affiliation:
Department of Computer-Science, Indian Institute of Technology, Madras 600036, India; (rangan@iitm.ernet.in)
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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.

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Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 34 , Issue 3 , May 2000, pp. 173 - 181
Copyright
© EDP Sciences, 2000

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