Streaming Algorithms for Network Design

TL;DR

The paper presents a general, streaming-friendly framework for Survivable Network Design Problems (SNDP) that leverages fault-tolerant spanners to compress the input while preserving feasibility. By coupling fault-tolerant spanners with LP-based fractional analyses, it achieves near-optimal space–approximation tradeoffs for VC-SNDP, EC-SNDP, and ELC-SNDP, including an $O(k^{1-1/t}n^{1+1/t})$ space target for VC-SNDP and an $O(t)$-approximation for EC-SNDP in near-linear space, among other refinements. The approach extends to non-uniform models and yields efficient streaming algorithms, with improvements over prior work such as JKMV24, and it also handles vertex-connectivity augmentation in the link-arrival model using SPQR-tree techniques to obtain constant-factor approximations for small k. A matching set of lower bounds shows that, in general, achieving better-than-$2t+1$-approximation in a single pass demands near-optimal space, underscoring the effectiveness and limits of the proposed framework. Overall, the work advances single-pass streaming network design by integrating spanner-based cores with LP insights and graph-decomposition tools, enabling scalable, provable-quality solutions for large streaming graphs.

Abstract

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 pair of $u, v \in V$, $u$ and $v$ are $r(uv)$-edge/vertex-connected. Recent work by Jin et al. [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 results 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 $O(βt)$-approximation where $β$ is the integrality gap of the natural cut-based LP relaxation. When applied to the EC-SNDP, our framework provides an $O(t)$-approximation in $\tilde O(k^{1/2-1/(2t)}n^{1 + 1/t} + kn)$ space, improving the $O(t \log k)$-approximation of [JKMV24] using $\tilde O(kn^{1+1/t})$ space; 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.

Figures (10)

• Figure 1: Biset-based LP relaxation of VC-SNDP
• Figure 2: Construction of the fractional solution $\boldsymbol{x}$ of \ref{['fig:VC-SNDP-LP']} from $\textnormal{OPT}$
• Figure 3: Examples for Lemma \ref{['lem:vc_1_to_2_feasibility']}. $a$ represents the deleted vertex and $b = LCA(u,v)$. Dashed lines represent paths while solid edges represent edges.
• Figure 4: Example of separation pair $\{a, b\}$ and corresponding separation classes shown with different colors.
• Figure 5: Example of an SPQR tree $T$ constructed from 2-connected graph $G$. Suppose we root the tree at node $C$. As defined in Section \ref{['sec:vc_2_to_3_algo']}, $\textnormal{parent}(E) = \{5, 6\}$, and $A, B, D$ all have parent $\{1, 2\}$ (corresponding to different virtual edges). In this example, $h(1) = h(2) = C$, but $\ell(1), \ell(2)$ can be chosen arbitrarily from $\{A, B, D\}$. Vertices $3$ and $4$ are not included in any virtual edges; thus $h(3) = \ell(3) = A$ and $h(4) = \ell(4) = B$. The same holds for vertices $7,8,9,10$; $h$ and $\ell$ both map to $E$. Finally, $h(5) = h(6) = D$ and $\ell(5) = \ell(6) = E$.

Theorems & Definitions (56)

• Remark 1.1
• Remark 1.2
• Theorem 2.1: Edge-connectivity Menger's theorem
• Theorem 2.2: Vertex-connectivity Menger's theorem
• Theorem 2.3: Element-connectivity Menger's theorem
• Definition 2.4: Fault-Tolerant Spanners
• Theorem 2.5: bodwin2019trivial
• Theorem 2.6: bodwin2022partially
• Theorem 2.7: Streaming Weighted VFT Spanners
• Theorem 2.8: Streaming Weighted EFT Spanners