The Steiner Tree Reoptimization Problem with Sharpened Triangle Inequality
- Cite this paper as:
- Böckenhauer HJ., Freiermuth K., Hromkovič J., Mömke T., Sprock A., Steffen B. (2010) The Steiner Tree Reoptimization Problem with Sharpened Triangle Inequality. In: Calamoneri T., Diaz J. (eds) Algorithms and Complexity. CIAC 2010. Lecture Notes in Computer Science, vol 6078. Springer, Berlin, Heidelberg
In this paper, we deal with several reoptimization variants of the Steiner tree problem in graphs obeying a sharpened β-triangle inequality. A reoptimization algorithm exploits the knowledge of an optimal solution to a problem instance for finding good solutions for a locally modified instance. We show that, in graphs satisfying a sharpened triangle inequality (and even in graphs where edge-costs are restricted to the values 1 and 1 + γ for an arbitrary small γ> 0), Steiner tree reoptimization still is NP-hard for several different types of local modifications, and even APX-hard for some of them.
As for the upper bounds, for some local modifications, we design linear-time (1/2 + β)-approximation algorithms, and even polynomial-time approximation schemes, whereas for metric graphs (β= 1), none of these reoptimization variants is known to permit a PTAS. As a building block for some of these algorithms, we employ a 2β-approximation algorithm for the classical Steiner tree problem on such instances, which might be of independent interest since it improves over the previously best known ratio for any β < 1/2 + ln (3)/4 ≈ 0.775.
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