2022/07/16 by Ravi, R., Zhang, Weizhong, Zlatin, Michael · 1 citation
#Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2207.07983
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~\citeJ2001. 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~\citeTZ2022 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~\citeN2010, there exists an O(log |R|)-approximation for this problem. We provide an O(log2 (|R|))-approximation algorithm for NW-STAP using a greedy algorithm leveraging the spider decomposition of optimal solutions.