A PTAS for Three-Edge Connectivity in Planar Graphs

TL;DR

This paper presents a PTAS for the minimum-weight subgraph problem with specified connectivity requirements in planar graphs, extending previous results to the case where vertices require up to three edge-disjoint paths.

Contribution

The paper introduces a PTAS for three-edge connectivity in planar graphs and establishes new properties of triconnected planar graphs that facilitate this approximation.

Findings

Successfully extended PTAS to three-edge connectivity case

Proved new properties of triconnected planar graphs

Achieved polynomial-time approximation scheme for the problem

Abstract

We consider the problem of finding the minimum-weight subgraph that satisfies given connectivity requirements. Specifically, given a requirement $r \in \{0,1,2,3\}$ for every vertex, we seek the minimum-weight subgraph that contains, for every pair of vertices $u$ and $v$, at least $\min\{ r(v), r(u)\}$ edge-disjoint $u$-to-$v$ paths. We give a polynomial-time approximation scheme (PTAS) for this problem when the input graph is planar and the subgraph may use multiple copies of any given edge. This generalizes an earlier result for $r \in \{0,1,2\}$. In order to achieve this PTAS, we prove some properties of triconnected planar graphs that may be of independent interest.