Improved Approximation for Two-Edge-Connectivity.

The basic goal of survivable network design is to construct low-cost networks which preserve a sufficient level of connectivity despite the failure or removal of a few nodes or edges. One of the most basic problems in this area is the $2$-Edge-Connected Spanning Subgraph problem (2-ECSS): given an undirected graph $G$, find a $2$-edge-connected spanning subgraph $H$ of $G$ with the minimum number of edges (in particular, $H$ remains connected after the removal of one arbitrary edge). 2-ECSS is NP-hard and the best-known (polynomial-time) approximation factor for this problem is $4/3$. Interestingly, this factor was achieved with drastically different techniques by [Hunkenschr{\"o}der, Vempala and Vetta '00,'19] and [Seb{\"o} and Vygen, '14]. In this paper we present an improved $\frac{118}{89}+\epsilon<1.326$ approximation for 2-ECSS. The key ingredient in our approach (which might also be helpful in future work) is a reduction to a special type of structured graphs: our reduction preserves approximation factors up to $6/5$. While reducing to 2-vertex-connected graphs is trivial (and heavily used in prior work), our structured graphs are "almost" 3-vertex-connected: more precisely, given any 2-vertex-cut $\{u,v\}$ of a structured graph $G=(V,E)$, $G[V\setminus \{u,v\}]$ has exactly 2 connected components, one of which contains exactly one node of degree $2$ in $G$.