A Unified Approach for Approximating 2-Edge-Connected Spanning Subgraph and 2-Vertex-Connected Spanning Subgraph

TL;DR

This paper presents a unified $4/3$-approximation approach for both $2$-edge-connected spanning subgraphs and $2$-vertex-connected spanning subgraphs. It starts from an inclusion-wise minimal $2$-VCSS and uses a recursive, local-improvement framework based on strong short segments, with a final cleanup for $2$-ECSS. The core contribution is a dual-cost-sharing analysis on the natural LP relaxations that proves $|F| \le \frac{4}{3}\mathrm{OPT}$, along with a reduction argument that preserves optimality bounds across transformed graphs. A tight example shows the bound is tight, and the unified method elegantly bridges the two problems with polynomial-time solvability.

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 both $\frac{4}{3}$. Using a common theme, the algorithms and their analyses are very similar.

Figures (15)

• Figure 1: An example of an improvement operation of size 1
• Figure 2: An example of an improvement operation of size 1
• Figure 3: An example of an improvement operation of size 1
• Figure 4: An example of an improvement operation of size 1
• Figure 5: An example of an improvement operation of size 2

Theorems & Definitions (16)

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