New and improved 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 $R \subseteq V$, and a Steiner tree $T$ spanning $R$. The edges $L := E \setminus E(T)$ 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 \cite{J2001}. We give the first polynomial time algorithm for STAP with approximation ratio better than 2. In particular we achieve a ratio of $(1+ \ln 2 + \varepsilon) \approx 1.69 + \varepsilon$. To do this, we use the Local Greedy approach of \cite{TZ2021} 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 $S \subseteq V \setminus R$ so that $G[R \cup S]$ is 2-edge-connected. We provide a $O(\log^2 (|R|))$-approximation algorithm for NW-STAP. To do this, we use a greedy algorithm leveraging the spider decomposition of optimal solutions.