We consider the Survivable Network Design problem (SNDP) in the single-pass insertion-only streaming model. The input to SNDP is an edge-weighted graph $G = (V, E)$ and an integer connectivity requirement $r(uv)$ for each $u, v \in V$. The objective is to find a min-weight subgraph $H \subseteq G$ s.t., for every $u, v \in V$, $u$ and $v$ are $r(uv)$-edge/vertex-connected. Recent work by [JKMV24] obtained approximation algorithms for edge-connectivity augmentation, and via that, also derived algorithms for edge-connectivity SNDP (EC-SNDP). We consider vertex-connectivity setting (VC-SNDP) and obtain several results for it as well as improved bounds for EC-SNDP. * We provide a general framework for solving connectivity problems in streaming; this is based on a connection to fault-tolerant spanners. For VC-SNDP we provide an $O(tk)$-approximation in $\tilde O(k^{1-1/t}n^{1 + 1/t})$ space, where $k$ is the maximum connectivity requirement, assuming an exact algorithm at the end of the stream. Using a refined LP-based analysis, we provide an $O(\beta t)$-approximation in polynomial time, where $\beta$ is the best polytime approximation w.r.t. the optimal fractional solution to a natural LP relaxation. When applied to EC-SNDP, our framework provides an $O(t)$-approximation in $\tilde O(k^{1-1/t}n^{1 + 1/t})$ space, improving the $O(t \log k)$-approximation of [JKMV24]; this also extends to element-connectivity SNDP. * We consider vertex connectivity-augmentation in the link-arrival model. The input is a $k$-vertex-connected subgraph $G$, and the weighted links $L$ arrive in the stream; the goal is to store the min-weight set of links s.t. $G \cup L$ is $(k+1)$-vertex-connected. We obtain $O(1)$ approximations in near-linear space for $k = 1, 2$. Our result for $k=2$ is based on SPQR tree, a novel application for this well-known representation of $2$-connected graphs.

翻译：本文在单遍插入式流模型中研究可生存网络设计问题（SNDP）。SNDP的输入是一个边赋权图$G = (V, E)$以及每对顶点$u, v \in V$的整数连通度需求$r(uv)$。目标是找到一个最小权重的子图$H \subseteq G$，使得对于任意$u, v \in V$，$u$和$v$之间具有$r(uv)$-边连通/顶点连通性。[JKMV24]的最新工作通过边连通度增强获得了近似算法，并由此推导出边连通SNDP（EC-SNDP）的算法。我们研究顶点连通场景（VC-SNDP），不仅为此获得多项结果，还改进了EC-SNDP的界。* 我们提出一个解决流式连通性问题的通用框架，该框架基于与容错生成子的关联。对于VC-SNDP，我们在$\tilde O(k^{1-1/t}n^{1 + 1/t})$空间内提供$O(tk)$近似解（假设流结束时采用精确算法），其中$k$为最大连通需求。通过基于线性规划的精细分析，我们在多项式时间内给出$O(\beta t)$近似解，其中$\beta$是针对自然线性规划松弛最优分数解的最佳多项式时间近似比。将该框架应用于EC-SNDP时，我们在$\tilde O(k^{1-1/t}n^{1 + 1/t})$空间内获得$O(t)$近似解，改进了[JKMV24]的$O(t \log k)$近似结果；该框架还可扩展至元素连通SNDP。* 我们在链接到达模型中研究顶点连通度增强问题。输入为$k$-顶点连通子图$G$，加权链接$L$以流式到达；目标是存储最小权重的链接集合，使得$G \cup L$达到$(k+1)$-顶点连通。对于$k = 1, 2$的情况，我们在近线性空间内获得$O(1)$近似解。其中$k=2$的结果基于SPQR树——这是对该经典2-连通图表示方法的新颖应用。