9/7-Approximation for Two-Edge-Connectivity and Two-Vertex-Connectivity

TL;DR

This work presents a polynomial-time $rac{9}{7}$-approximation algorithm for both the $2$-edge-connected spanning subgraph problem ($2$-ECSS) and the $2$-vertex-connected spanning subgraph problem ($2$-VCSS). The approach starts from an inclusion-wise minimal $2$-VCSS and repeatedly applies carefully designed improvement operations on strong short segments, supported by a recursive enhancement and a final cleanup, all analyzed through a dual-fitting cost-sharing framework. The analysis yields a bound of $rac{9}{7}$ relative to the optimum, supported by a tight example that matches the bound. This unified method advances the frontier beyond prior ratios around $rac{4}{3}$ and $rac{3}{2}$ and provides a robust, practically relevant guarantee for network design problems involving 2-connectivity.

Abstract

We provide algorithms for the minimum 2-edge-connected spanning subgraph problem and the minimum 2-vertex-connected spanning subgraph problem with approximation ratio $\frac{9}{7}$. This improves upon a recent algorithm with ratio slightly smaller than $\frac{4}{3}$ for 2-edge-connectivity, and another one with ratio $\frac{4}{3}$ for 2-vertex-connectivity.

Figures (22)

• Figure 1: An example of an improvement operation on a strong 2-segment
• Figure 2: An example of an improvement operation on a strong 2-segment
• Figure 3: An example of an improvement operation on a strong 3-segment
• Figure 4: An example of an improvement operation on a strong 4-segment
• Figure 5: An example of an improvement process of recursion depth $2$

Theorems & Definitions (19)

• Theorem 1
• Proposition 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Claim 5
• proof
• Claim 6