A Better-Than-$5/4$-Approximation for Two-Edge Connectivity

The 2-Edge-Connected Spanning Subgraph Problem (2ECSS) is a fundamental problem in survivable network design. Given an undirected $2$-edge-connected graph, the goal is to find a $2$-edge-connected spanning subgraph with the minimum number of edges; a graph is 2-edge-connected if it is connected after the removal of any single edge. 2ECSS is APX-hard and has been extensively studied in the context of approximation algorithms. Very recently, Bosch-Calvo, Garg, Grandoni, Hommelsheim, Jabal Ameli, and Lindermayr showed the currently best-known approximation ratio of $\frac{5}{4}$ [STOC 2025]. This factor is tight for many of their techniques and arguments, and it was not clear whether $\frac{5}{4}$ can be improved. We break this natural barrier and present a $(\frac{5}{4} - η)$-approximation algorithm, for some constant $η\geq 10^{-6}$. On a high level, we follow the approach of previous works: take a triangle-free $2$-edge cover and transform it into a 2-edge-connected spanning subgraph by adding only a few additional edges. For $\geq \frac{5}{4}$-approximations, one can heavily exploit that a $4$-cycle in the 2-edge cover can ``buy'' one additional edge. This enables simple and nice techniques, but immediately fails for our improved approximation ratio. To overcome this, we design two complementary algorithms that perform well for different scenarios: one for few $4$-cycles and one for many $4$-cycles. Besides this, there appear more obstructions when breaching $\frac54$, which we surpass via new techniques such as colorful bridge covering, rich vertices, and branching gluing paths.