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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