A $4/3$ ratio approximation algorithm for the Tree Augmentation Problem by deferred local-ratio and climbing

The \emph{Tree Augmentation Problem (TAP)} is given a tree $T=(V,E_T)$ and additional set of {\em links} $E$ on $V\times V$, find $F \subseteq E$ such that $T \cup F$ is $2$-edge-connected, and $|F|$ is minimum. The problem is APX-hard \cite{r} even in if links are only between leaves \cite{r}. The best known approximation ratio for TAP is $1.393$, due to Traub and Zenklusen~\cite{tr1} J.~ACM,~2025 using the {\em relative greedy} technique \cite{zel}. \noindent We introduce a new technique called the {\em deferred local ratio technique}. In this technique, the disjointness of the local-ratio primal-dual type does not hold. The technique applies Set Cover problem under certain conditions (see Section \ref{lr}). We use it provide a We use it to provide a $4/3$ approximation algorithm for TAP. It is possible this technique will find future applications. The running time is The running time is $O(m\cdot\sqrt{n})$ time \cite{vaz}, \cite{vaz1}. Faster than \cite{tr1} \cite{LS} and LP based algorithms as we do not enumeratestructures of size $exp(\Theta(f(1/\epsilon)\cdot \log n)).$ Nor do we scale and round. \noindent \cite{ed} has an implementation \cite{kol} that is extensively used in the industry.