On the stability of approximation for Hamiltonian path problems

On the stability of approximation for Hamiltonian path problems

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Article ID: iaor20062928
Country: Canada
Volume: 1
Issue: 1
Start Page Number: 31
End Page Number: 45
Publication Date: Jan 2006
Journal: Algorithmic Operations Research
Authors: , , ,
Keywords: graphs
Abstract:

We consider the problem of finding a cheapest Hamiltonian path of a complete graph satisfying a relaxed triangle inequality, i.e., such that for some parameter β > 1, the edge costs satisfy the inequality c({x,y}) ≤ β(c({x,z}) + c({z,y})) for every triple of vertices x, y, z. There are three variants of this problem, depending on the number of prespecified endpoints: zero, one, or two. For metric graphs there exist approximation algorithms, with approximation ratio 3/2 for the first two variants and 5/3 for the latter one. Using results on the approximability of the Travelling Salesman Problem with input graphs satisfying the relaxed triangle inequality, we obtain for our problem approximation algorithms with ratio min2 + β,3/2β2) for zero or one prespecified endpoints, and 5/3 β2 for two endpoints.

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