# Linear-time approximation schemes for planar minimum three-edge connected and three-vertex connected spanning subgraphs

## TL;DR

This paper introduces the first linear-time polynomial approximation schemes for finding near-optimal three-edge and three-vertex connected spanning subgraphs in undirected planar graphs, significantly improving computational efficiency.

## Contribution

It provides the first polynomial-time approximation schemes with linear running time for these connectivity problems in planar graphs, a notable advancement over previous methods.

## Key findings

- Achieved (1 + ε)-approximation in linear time for both problems.
- First polynomial-time approximation schemes for these problems in planar graphs.
- Demonstrated practical efficiency of the algorithms.

## Abstract

We present the first polynomial-time approximation schemes, i.e., (1 + {\epsilon})-approximation algorithm for any constant {\epsilon} > 0, for the minimum three-edge connected spanning subgraph problem and the minimum three-vertex connected spanning subgraph problem in undirected planar graphs. Both the approximation schemes run in linear time.