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

2025-09-23

AI Summary

This paper studies the 2-Edge-Connected Spanning Subgraph problem (2ECSS): given an undirected 2-edge-connected graph, find a 2-edge-connected spanning subgraph with the minimum number of edges. A core problem in survivable network design, 2ECSS had long resisted improvement beyond the best-known 5/4-approximation ratio. We break this barrier for the first time, presenting the first (5/4 − η)-approximation algorithm for some constant η ≥ 10⁻⁶. Our approach introduces novel combinatorial structures—colored bridge covers, rich vertices, and branching-merging paths—and builds upon a triangle-free 2-edge-cover framework. To handle 4-cycles, we devise a pair of complementary algorithms whose analyses are tightly coupled. The method integrates advanced combinatorial optimization techniques with refined graph-theoretic analysis. This result constitutes a substantial theoretical advance and establishes a new paradigm and technical toolkit for designing approximation algorithms for fault-tolerant network design problems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

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 $ηgeq 10^{-6}$. 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.

Problem

• Finding minimum 2-edge-connected spanning subgraphs in undirected graphs
• Breaking the 5/4 approximation barrier for the 2ECSS problem
• Developing new techniques to overcome limitations of previous approaches

Innovation

• Developed two complementary algorithms for different scenarios
• Introduced colorful bridge covering technique for connectivity
• Utilized rich vertices and branching gluing paths methods