Reoptimization of Steiner trees: Changing the terminal set

Given an instance of the Steiner tree problem together with an optimal solution, we consider the scenario where this instance is modified locally by adding one of the vertices to the terminal set or removing one vertex from it. In this paper, we investigate the problem whether the knowledge of an optimal solution to the unaltered instance can help in solving the locally modified instance. Our results are as follows: (i) We prove that these reoptimization variants of the Steiner tree problem are NP-hard, even if edge costs are restricted to values from {1, 2}. (ii) We design 1.5-approximation algorithms for both variants of local modifications. This is an improvement over the currently best known approximation algorithm for the classical Steiner tree problem which achieves an approximation ratio of 1+ln(3)/2 ≈ 1.55. (iii) We present a PTAS for the subproblem in which the edge costs are natural numbers {1, . . . , k} for some constant k. ⋆ This work was partially supported by SBF grant C 06.0108 as part of the COST 293 (GRAAL) project funded by the European Union. ∗ Corresponding author. Address: Department of Computer Science, ETH Zurich, Universitätsstrasse 6, 8092 Zurich, Switzerland. Email addresses: [email protected] (Hans-Joachim Böckenhauer), [email protected] (Juraj Hromkovič), [email protected] (Richard Královič), [email protected] (Tobias Mömke), [email protected] (Peter Rossmanith). 1 This author was visiting ETH Zurich while part of this work was done.