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 $exp(Θ(f(1/ε)\cdot \log n)).$ Nor do we scale and round. \noindent \cite{ed} has an implementation \cite{kol} that is extensively used in the industry.

翻译：\emph{树增广问题（Tree Augmentation Problem，TAP）}的定义如下：给定一棵树$T=(V,E_T)$以及顶点集$V$上的附加\emph{边链}集合$E$，目标是找到$F \subseteq E$，使得$T \cup F$是$2$-边连通的，且$|F|$最小。即使边链仅存在于叶子节点之间，该问题也是APX难的\cite{r}。目前TAP的最佳已知近似比为$1.393$，由Traub和Zenklusen~\cite{tr1}（J.~ACM,~2025）利用\emph{相对贪心}技术\cite{zel}获得。\noindent 本文提出一种称为\emph{延迟局部比率技术}的新方法。在该技术中，局部比率原始-对偶类型的不相交性不再成立。该技术可在特定条件下应用于集合覆盖问题（参见第\ref{lr}节）。我们利用该技术为TAP设计了一个$4/3$近似算法。此技术未来可能在其他问题中得到应用。算法运行时间为$O(m\cdot\sqrt{n})$ \cite{vaz}, \cite{vaz1}。相较于\cite{tr1}、\cite{LS}以及基于线性规划的算法，本方法速度更快，因为我们无需枚举规模为$exp(Θ(f(1/ε)\cdot \log n))$的结构，也无需进行缩放与舍入操作。\noindent 文献\cite{ed}中提供了一个实现版本\cite{kol}，该版本已在工业界得到广泛应用。