The Weighted Connectivity Augmentation Problem is the problem of augmenting the edge-connectivity of a given graph by adding links of minimum total cost. This work focuses on connectivity augmentation problems in the Steiner setting, where we are not interested in the connectivity between all nodes of the graph, but only the connectivity between a specified subset of terminals. We consider two related settings. 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 and nodes from $G$ of minimum cost so that the edge-connectivity between nodes of $H$ increases by 1. 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. All of the above problems can be approximated to within a factor of 2 using e.g. Jain's iterative rounding algorithm for Survivable Network Design. In this work, we leverage the framework of Traub and Zenklusen to give a $(1 + \ln{2} +\varepsilon)$-approximation for the Steiner Ring Augmentation Problem (SRAP): given a cycle $H = (V(H),E)$ embedded in a larger graph $G = (V, E \cup L)$ and a subset of terminals $R \subseteq V(H)$, choose a subset of links $S \subseteq L$ of minimum cost so that $(V, E \cup S)$ has 3 pairwise edge-disjoint paths between every pair of terminals. We show this yields a polynomial time algorithm with approximation ratio $(1 + \ln{2} + \varepsilon)$ for $2$-SCAP. We obtain an improved approximation guarantee of $(1.5+\varepsilon)$ for SRAP in the case that $R = V(H)$, which yields a $(1.5+\varepsilon)$-approximation for $k$-SAG for any $k$.

翻译：加权连通性增强问题是指通过添加最小总成本的链路来增强给定图的边连通性。本文关注Steiner设置下的连通性增强问题，在该设置中，我们不仅关心图中所有节点之间的连通性，而只关心指定终端子集之间的连通性。我们考虑两种相关设置。在图的Steiner增强问题（$k$-SAG）中，给定图$G$的一个$k$边连通子图$H$，目标是通过以最小成本包含$G$中的链路和节点来增强$H$，使得$H$节点之间的边连通性增加1。在Steiner连通性增强问题（$k$-SCAP）中，给定一个连接终端$R$的Steiner $k$边连通图，目标是添加最小成本的链路以创建$R$的Steiner $(k+1)$边连通图。注意，$k$-SAG是$k$-SCAP的特例。所有上述问题均可通过例如Jain针对可生存网络设计的迭代取整算法达到2倍以内的近似比。本文利用Traub和Zenklusen的框架，为Steiner环增强问题（SRAP）提出$(1 + \ln{2} +\varepsilon)$近似算法：给定嵌入更大图$G = (V, E \cup L)$中的环$H = (V(H),E)$及终端子集$R \subseteq V(H)$，选择最小成本的链路子集$S \subseteq L$，使得$(V, E \cup S)$中每对终端之间存在3条两两边不相交的路径。我们证明，由此可推导出$2$-SCAP的多项式时间算法，近似比为$(1 + \ln{2} + \varepsilon)$。当$R = V(H)$时，我们获得了SRAP的改进近似保证$(1.5+\varepsilon)$，进而对任意$k$的$k$-SAG问题实现$(1.5+\varepsilon)$近似。