Approximation Algorithms for Steiner Tree Augmentation Problems

In the Steiner Tree Augmentation Problem (STAP), we are given a graph G = (V, E), a set of terminals R ⊆ V, and a Steiner tree T spanning R. The edges L: = E\E(T) are called links and have non-negative costs. The goal is to augment T by adding a minimum cost set of links, so that there are 2 edge-disjoint paths between each pair of vertices in R. This problem is a special case of the Survivable Network Design Problem, which can be approximated to within a factor of 2 using iterative rounding [13]. We give the first polynomial time algorithm for STAP with approximation ratio better than 2. In particular, we achieve an approximation ratio of (1. 5 + ε). To do this, we employ the Local Search approach of [24] for the Tree Augmentation Problem and generalize their main decomposition theorem from links (of size two) to hyper-links. We also consider the Node-Weighted Steiner Tree Augmentation Problem (NW-STAP) in which the non-terminal nodes have non-negative costs. We seek a cheapest subset S ⊆ V\R so that G[R ∪ S ] is 2-edge-connected. Using a result of Nutov [18], there exists an O (log | R |)-approximation for this problem. We provide an O (log 2 (| R |))-approximation algorithm for NW-STAP using a greedy algorithm leveraging the spider decomposition of optimal solutions.

Authors

• R. Ravi 0001
• Weizhong Zhang
• Michael Zlatin

Keywords

No keywords are indexed for this paper.