The $2$-Edge-Connected Spanning Subgraph problem (2-ECSS) is one of the most fundamental and well-studied problems in the context of network design. In the problem, we are given an undirected graph $G$, and the objective is to find a $2$-edge-connected spanning subgraph $H$ of $G$ with the minimum number of edges. For this problem, a lot of approximation algorithms have been proposed in the literature. In particular, very recently, Garg, Grandoni, and Ameli gave an approximation algorithm for 2-ECSS with factor $1.326$, which was the best approximation ratio. In this paper, we give a $(1.3+\varepsilon)$-approximation algorithm for 2-ECSS, where $\varepsilon$ is an arbitrary positive fixed constant, which improves the previously known best approximation ratio. In our algorithm, we compute a minimum triangle-free $2$-edge-cover in $G$ with the aid of the algorithm for finding a maximum triangle-free $2$-matching given by Hartvigsen. Then, with the obtained triangle-free $2$-edge-cover, we apply the arguments by Garg, Grandoni, and Ameli.

翻译：$2$-连通支撑子图问题（2-ECSS）是网络设计中最基础且被广泛研究的问题之一。在该问题中，给定一个无向图 $G$，目标是找到 $G$ 的一个边数最少的 $2$-边连通支撑子图 $H$。针对该问题，文献中已提出大量近似算法。特别地，最近Garg、Grandoni和Ameli给出了因子为 $1.326$ 的2-ECSS近似算法，这是当时的最佳近似比。在本文中，我们提出了一个 $(1.3+\varepsilon)$-近似算法用于2-ECSS问题，其中 $\varepsilon$ 为任意正常数，该算法改进了此前已知的最佳近似比。在我们的算法中，借助Hartvigsen提出的最大无三角形 $2$-匹配算法，计算 $G$ 中的最小无三角形 $2$-边覆盖。随后，利用所得的无三角形 $2$-边覆盖，应用Garg、Grandoni和Ameli的论证方法。