Two-Edge Connectivity via Pac-Man Gluing

TL;DR

This paper presents a polynomial-time approximation algorithm for the NP-hard 2-edge-connected spanning subgraph problem, achieving a ratio of 5/4 + ε, by leveraging high vertex connectivity and a Pac-Man inspired gluing technique.

Contribution

It introduces a novel approach that reduces the problem to structured graphs with high vertex connectivity and employs a Pac-Man inspired gluing method, improving approximation ratio and simplifying analysis.

Findings

Achieves a 5/4 + ε approximation ratio for 2-ECSS.

Introduces a Pac-Man inspired gluing technique for merging components.

Simplifies the algorithm and analysis compared to previous methods.

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.