The steiner problem with edge lengths 1 and 2,
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Reoptimization of Steiner Trees
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In this paper, we deal with several reoptimization variants of the Steiner tree problem in graphs obeying a sharpened @b-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+@c for an arbitrary small @c0), 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+@b)-approximation algorithms, and even polynomial-time approximation schemes, whereas for metric graphs (@b=1), none of these reoptimization variants is known to permit a PTAS. As a building block for some of these algorithms, we employ a 2@b-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 @b