A simplified 1.5-approximation algorithm for augmenting edge-connectivity of a graph from 1 to 2

TL;DR

This paper presents a new, simplified, and self-contained 1.5-approximation algorithm for the Tree Augmentation Problem, improving the clarity and correctness of previous complex proofs.

Contribution

Provides a correct, shorter, and more accessible proof of a 1.5-approximation algorithm for TAP, resolving previous errors and complexities.

Findings

Achieved a 1.5-approximation ratio for TAP

Simplified the proof significantly compared to previous versions

Confirmed the correctness of the algorithm

Abstract

The Tree Augmentation Problem (TAP) is: given a connected graph $G=(V,{\cal E})$ and an edge set $E$ on $V$ find a minimum size subset of edges $F \subseteq E$ such that $(V,{\cal E} \cup F)$ is $2$-edge-connected. In the conference version \cite{EFKN-APPROX} was sketched a $1.5$-approximation algorithm for the problem. Since a full proof was very complex and long, the journal version was cut into two parts. In the first part \cite{EFKN-TALG} was only proved ratio $1.8$. An attempt to simplify the second part produced an error in \cite{EKN-IPL}. Here we give a correct, different, and self contained proof of the ratio $1.5$, that is also substantially simpler and shorter than the previous proofs.