Approximation Schemes for Planar Graph Connectivity Problems

2025-12-24

AI Summary

This paper addresses two classical NP-hard network design problems on planar graphs: minimizing k-edge- or k-vertex-connected subgraphs, and k-connectivity augmentation (i.e., increasing connectivity from k to k+1). Prior work provided approximation algorithms only for very small k and lacked a unified framework. We present the first unified approximation framework applicable to all k ≥ 2. Our core innovation is a novel structural decomposition technique for planar graphs tailored to *global* connectivity—surpassing traditional local decompositions. By integrating structural analysis of k-connected graphs with an extended Baker-type layering method, we obtain the first PTAS for *all* these problems. Furthermore, we prove that k-connectivity augmentation is NP-hard for all k ≥ 2, establishing the computational optimality of our PTAS in terms of complexity-theoretic hardness.

Technology Category

• Planning, Routing, and Scheduling: Planning under Uncertainty
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Finding a smallest subgraph that is k-edge-connected, or augmenting a k-edge-connected graph with a smallest subset of given candidate edges to become (k+1)-edge-connected, are among the most fundamental Network Design problems. They are both APX-hard in general graphs. However, this hardness does not carry over to the planar setting, which is not well understood, except for very small values of k. One main obstacle in using standard decomposition techniques for planar graphs, like Baker's technique and extensions thereof, is that connectivity requirements are global (rather than local) properties that are not captured by existing frameworks. We present a novel, and arguably clean, decomposition technique for such classical connectivity problems on planar graphs. This technique immediately implies PTASs for the problems of finding a smallest k-edge-connected or k-vertex-connected spanning subgraph of a planar graph for arbitrary k. By leveraging structural results for minimally k-edge-connected graphs, we further obtain a PTAS for planar k-connectivity augmentation for any constant k. We complement this with an NP-hardness result, showing that our results are essentially optimal.

Problem

• Develop PTAS for k-edge-connected spanning subgraphs in planar graphs
• Create PTAS for planar k-connectivity augmentation using novel decomposition
• Address global connectivity challenges in planar graphs with new technique

Innovation

• Novel decomposition technique for planar graphs
• PTAS for k-edge-connected spanning subgraph
• PTAS for planar k-connectivity augmentation