Streaming Algorithms for Network Design

2025-03-02

AI Summary

This paper studies the Survivable Network Design Problem (SNDP) in the single-insertion dynamic graph stream model, addressing both vertex-connectivity (VC-SNDP) and edge-connectivity (EC-SNDP) variants, with the goal of maintaining a minimum-weight subgraph satisfying given connectivity requirements under edge arrivals. We establish the first theoretical connection between SNDP and fault-tolerant spanners, and propose a unified streaming algorithmic framework based on LP relaxation analysis, SPQR-tree structural decomposition, and streaming sampling techniques. Our key contributions are: (i) an $O(t)$-approximation for EC-SNDP—improving upon the prior $O(t log k)$ bound; (ii) an $O(eta t)$-approximation for VC-SNDP in polynomial time, where $eta$ is the approximation ratio of the underlying Steiner Forest subroutine; and (iii) the first $O(1)$-approximation algorithms with near-linear space for $k$-vertex-connectivity augmentation for $k=1,2$. All results are achieved in a single-pass streaming setting, balancing theoretical optimality and practical efficiency.

Technology Category

• Data Mining & Knowledge Management: Data Stream Mining
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $u, v in V$, $u$ 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 $ ilde 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 $O(eta t)$-approximation in polynomial time, where $eta$ 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 $ ilde 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.

Problem

• Develops streaming algorithms for Survivable Network Design (SNDP).
• Improves approximation bounds for edge-connectivity SNDP (EC-SNDP).
• Provides novel results for vertex-connectivity SNDP (VC-SNDP).

Innovation

• General framework for streaming connectivity problems
• O(tk)-approximation for VC-SNDP in streaming
• O(1) approximations for vertex connectivity-augmentation