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 nonterminal nodes have non-negative costs. We seek a cheapest subset S ⊆ V R so that G[R ∪ S] is 2edge-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.