Abstract
In this paper, we study the asymmetric traveling salesman problem (ATSP) with strengthened triangle inequality, i.e. for some \(\gamma\in [\frac{1}{2},1)\) the edge weights satisfy w(u,v) ≤ γ(w(u,x) + w(x,v)) for all distinct vertices u,v,x.
We present two approximation algorithms for this problem. The first one is very simple and has approximation ratio \(\frac{1}{2(1-\gamma)}\), which is better than all previous results for all \(\gamma \in (\frac{1}{2},1)\). The second algorithm is more involved but it also gives a much better approximation ratio: \(\frac{2-\gamma}{3(1-\gamma)}+O(\frac{1}{n})\) when γ > γ 0, and \(\frac{1}{2}(1+\gamma)^2 + \epsilon\) for any ε> 0 when γ ≤ γ 0, where γ 0 ≈ 0.7003.
This research is partially supported by a grant from the Polish Ministry of Science and Higher Education, project N206 005 32/0807.
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Kowalik, Ł., Mucha, M. (2009). Two Approximation Algorithms for ATSP with Strengthened Triangle Inequality. In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_41
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DOI: https://doi.org/10.1007/978-3-642-03367-4_41
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