Two-Edge Connectivity via Pac-Man Gluing

TL;DR

The paper advances the 2-ECSS problem by achieving a deterministic polynomial-time approximation of $5/4 + \varepsilon$, improving prior bounds. It introduces a modular framework: reduce arbitrary inputs to structured graphs, compute a triangle-free 2-edge cover, bridge to a bridgeless canonical cover, and glue components using a new 4-matching lemma in a Pac-Man inspired process. A key novelty is separating the gluing into two modular tasks (a preprocessing step to remove large 3-vertex cuts and a Pac-Man-like iterative gluing), which simplifies analysis and yields a clean approximation bound. The approach tightly interweaves combinatorial structure with careful credit accounting to bound the extra edges required during gluing, and it situates well within the broader landscape of network design and graph-TSP techniques.

Abstract

We study the 2-edge-connected spanning subgraph (2-ECSS) problem: Given a graph $G$, compute a connected subgraph $H$ of $G$ with the minimum number of edges such that $H$ is spanning, i.e., $V(H) = V(G)$, and $H$ is 2-edge-connected, i.e., $H$ remains connected upon the deletion of any single edge, if such an $H$ exists. The $2$-ECSS problem is known to be NP-hard. In this work, we provide a polynomial-time $(\frac 5 4 + \varepsilon)$-approximation for the problem for an arbitrarily small $\varepsilon>0$, improving the previous best approximation ratio of $\frac{13}{10}+\varepsilon$. Our improvement is based on two main innovations: First, we reduce solving the problem on general graphs to solving it on structured graphs with high vertex connectivity. This high vertex connectivity ensures the existence of a 4-matching across any bipartition of the vertex set with at least 10 vertices in each part. Second, we exploit this property in a later gluing step, where isolated 2-edge-connected components need to be merged without adding too many edges. Using the 4-matching property, we can repeatedly glue a huge component (containing at least 10 vertices) to other components. This step is reminiscent of the Pac-Man game, where a Pac-Man (a huge component) consumes all the dots (other components) as it moves through a maze. These two innovations lead to a significantly simpler algorithm and analysis for the gluing step compared to the previous best approximation algorithm, which required a long and tedious case analysis.

Figures (1)

• Figure 1: Illustrations of different cases considered in the proof of \ref{['lem:gluing:trivial-segment']}. Nodes of $\bm\hat{G}\xspace_H$ are depicted by a circle with black border, and vertices of $G$ are depicted by a black, filled circle. Edges of $H$ are black and solid, edges of $G \setminus H$ are gray and dashed, edges of $G \setminus H$ that are added to $H$ to obtain $H'$ are red, and edges that are removed from $H$ to obtain $H'$ are green.

Theorems & Definitions (38)

• Theorem 1
• Definition 1: $\alpha$-contractible subgraph GargGA23improved
• Definition 2: irrelevant edge GargGA23improved
• Definition 3: non-isolating 2-vertex cut GargGA23improved
• Definition 4: small and large 3-vertex cuts
• Definition 5: ($\alpha, \varepsilon)$-structured graph
• Lemma 1
• Lemma 2: $3$-Matching Lemma GargGA23improved
• Lemma 3: 4-Matching Lemma
• proof