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Analysis of a near-metric TSP approximation algorithm∗

Published online by Cambridge University Press:  06 August 2013

Sacha Krug
Sacha Krug*
Affiliation:
Department of Computer Science, ETH Zurich, Switzerland.. sacha.krug@inf.ethz.ch
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Abstract

The traveling salesman problem (TSP) is one of the most fundamental optimizationproblems. We consider the β-metric traveling salesman problem(Δβ-TSP), i.e., the TSPrestricted to graphs satisfying the β-triangle inequalityc({v,w}) ≤ β(c({v,u}) + c({u,w})),for some cost function c and any three vertices u,v,w.The well-known path matching Christofides algorithm (PMCA) guarantees an approximationratio of 3β2/2 and is the best known algorithm for theΔβ-TSP, for 1 ≤ β ≤ 2. Weprovide a complete analysis of the algorithm. First, we correct an error in the originalimplementation that may produce an invalid solution. Using a worst-case example, we thenshow that the algorithm cannot guarantee a better approximation ratio. The example canalso be used for the PMCA variants for the Hamiltonian path problem with zero and oneprespecified endpoints. For two prespecified endpoints, we cannot reuse the example, butwe construct another worst-case example to show the optimality of the analysis also inthis case.

Keywords

traveling salesman problemcombinatorial optimizationapproximation algorithmsgraph theory
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 47 , Issue 3 , July 2013 , pp. 293 - 314
Copyright
© EDP Sciences 2013

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References

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