Approximation Algorithms for Steiner Connectivity Augmentation

TL;DR

The paper advances Steiner connectivity augmentation by introducing SRAP as a central reduction and proving a $(1 + \ln{2} + \varepsilon)$-approximation for SRAP, which in turn yields improved guarantees for $2$-SCAP and $k$-SAG when $R=V(H)$. It extends the relative-greedy framework to the Steiner setting, using complete-instance preprocessing, $R$-special directed solutions, and a decomposition/DP-based optimization to manage hyper-links, enabling a $(1.5+\varepsilon)$-approximation for SRAP in the terminals-equals-ring case and hence a $(1.5+\varepsilon)$-approximation for $k$-SAG. The work coherently ties SRAP to classical WCAP/STAP via cactus representations and zero-cost link extensions, and provides a rigorous local-search strategy that achieves the improved bound under the $R=V(H)$ regime. Overall, the results push below the longstanding barrier of 2 for Steiner augmentation problems and offer a scalable, structured methodology with practical implications for designing robust Steiner networks.

Abstract

We consider connectivity augmentation problems in the Steiner setting, where the goal is to augment the edge-connectivity between a specified subset of terminal nodes. In the Steiner Augmentation of a Graph problem ($k$-SAG), we are given a $k$-edge-connected subgraph $H$ of a graph $G$. The goal is to augment $H$ by including links from $G$ of minimum cost so that the edge-connectivity between nodes of $H$ increases by 1. This is a generalization of the Weighted Connectivity Augmentation Problem, in which only links between pairs of nodes in $H$ are available for the augmentation. In the Steiner Connectivity Augmentation Problem ($k$-SCAP), we are given a Steiner $k$-edge-connected graph connecting terminals $R$, and we seek to add links of minimum cost to create a Steiner $(k+1)$-edge-connected graph for $R$. Note that $k$-SAG is a special case of $k$-SCAP. The results of Ravi, Zhang and Zlatin for the Steiner Tree Augmentation problem yield a $(1.5+\varepsilon)$-approximation for $1$-SCAP and for $k$-SAG when $k$ is odd (SODA'23). In this work, we give a $(1 + \ln{2} +\varepsilon)$-approximation for the Steiner Ring Augmentation Problem (SRAP). This yields a polynomial time algorithm with approximation ratio $(1 + \ln{2} + \varepsilon)$ for $2$-SCAP. We obtain an improved approximation guarantee for SRAP when the ring consists of only terminals, yielding a $(1.5+\varepsilon)$-approximation for $k$-SAG for any $k$.

Figures (8)

• Figure 1: We seek to augment the edge-connectivity between the grey nodes (terminals) in the given tree from 1 to 2. Nodes $a$, $b$ and $c$ are arranged in an equilateral unit triangle. If only direct links are allowed, the cheapest augmentation has cost 2, but utilizing a Steiner node in the center of this triangle allows us to establish 2-edge-connectivity between the terminals at a cost of $\sqrt{3} \approx 1.73$.
• Figure 2: The given graph has 2-edge-connectivity between the large grey nodes (the terminals), which we seek to increase to 3-edge-connectivity. Solving this with global connectivity augmentation results in a solution with 4 links when only 1 is required.
• Figure 3: A 3-SAG instance is shown on the left, and a 3-SCAP instance is shown on the right. The shaded nodes are terminals $R$, the black edges denote the edges of $E$ and the dashed edges represent the links $L$. In both pictures, the blue dashed links form a feasible solution.
• Figure 4: A SRAP instance where the black edges denote the given cycle, the dashed edges are the links, and the blue links form a feasible solution. The shaded nodes are the terminals $R$.
• Figure 5: An example of a SRAP instance undergoing preprocessing steps to obtain a complete instance. The leftmost SRAP instance has two undirected links $\ell_1$ (green) and $\ell_2$ (blue) in $L$. There are no links added in $L^1$. The middle picture shows the directed links added in $L^2$, where the green arcs are shadows of $\ell_1$ and have cost $c(\ell_1)$, and the blue arcs are shadows of the $\ell_2$ with cost $c(\ell_2)$. Finally, the third picture shows the (undominated) directed links added in $L^3$ in yellow. Each of these arcs have cost $c(\ell_1) + c(\ell_2)$. The final completed SRAP instance contains all of these links.

Theorems & Definitions (83)

• Lemma 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Definition 1.6
• Definition 1.7
• Definition 1.8
• Lemma 1.9
• proof
• Definition 1.10