The Steiner Tree Reoptimization Problem with Sharpened Triangle Inequality
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.
KeywordsMinimum Span Tree Steiner Tree Steiner Tree Problem Terminal Vertex Edge Cost
Unable to display preview. Download preview PDF.
- 2.Archetti, C., Bertazzi, L., Speranza, M.G.: Reoptimizing the 0-1 knapsack problem. Tech. Rep. 267, University of Brescia (2006)Google Scholar
- 8.Böckenhauer, H.J., Forlizzi, L., Hromkovič, J., Kneis, J., Kupke, J., Proietti, G., Widmayer, P.: Reusing optimal TSP solutions for locally modified input instances (extended abstract). In: Navarro, G., Bertossi, L.E., Kohayakawa, Y. (eds.) Proc. of the 4th IFIP International Conference on Theoretical Computer Science (TCS 2006). IFIP., vol. 209, pp. 251–270. Springer, New York (2006)CrossRefGoogle Scholar
- 16.Hwang, F.K., Richards, D.S., Winter, P.: The Steiner Tree Problems. In: Annals of Discrete Mathematics, vol. 53. North-Holland, Amsterdam (1992)Google Scholar
- 18.Prömel, H.J., Steger, A.: The Steiner Tree Problem. In: Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig (2002)Google Scholar
- 19.Robins, G., Zelikovsky, A.Z.: Improved Steiner tree approximation in graphs. In: Proc. of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2000), pp. 770–779. ACM/SIAM, New York (2000)Google Scholar
- 21.Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2004)Google Scholar