Approximation Schemes for Planar Graph Connectivity Problems

TL;DR

This paper introduces a new decomposition method for planar graphs that enables PTAS algorithms for classical connectivity problems, significantly advancing the understanding of these problems in the planar setting.

Contribution

The paper presents a novel decomposition technique for planar graphs that leads to PTASs for k-edge and k-vertex connectivity problems, extending the applicability beyond small k values.

Findings

PTASs for k-edge and k-vertex connectivity in planar graphs for any k

A new decomposition technique tailored for connectivity problems in planar graphs

NP-hardness results showing the limits of approximation for these problems

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.