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

TL;DR

The paper addresses the NP-hard problem of finding a minimum-size $2$-edge-connected spanning subgraph ($2$ECSS) and achieves a deterministic $5/4$-approximation in polynomial time, improving over the prior best $1.3+\

Abstract

The 2-Edge-Connected Spanning Subgraph problem (2ECSS) is among the most basic survivable network design problems: given an undirected and unweighted graph, the task is to find a spanning subgraph with the minimum number of edges that is 2-edge-connected (i.e., it remains connected after the removal of any single edge). 2ECSS is an NP-hard problem that has been extensively studied in the context of approximation algorithms. The best known approximation ratio for 2ECSS prior to this work was $1.3+\varepsilon$, for any constant $\varepsilon>0$ [Garg, Grandoni, Jabal-Ameli'23; Kobayashi, Noguchi'23]. In this paper, we present a 5/4-approximation algorithm. Our algorithm is also faster for small values of $\varepsilon$: its running time is $n^{O(1)}$ instead of $n^{O(1/\varepsilon)}$.

Figures (10)

• Figure 1: This figure is similar to figures in BGJ23 and GGJ23soda, where the authors also use similar notions of $\alpha$-structured graphs. The subgraph induced by the blue edges is a $5/4$-contractible graph. The red and green (resp. gray) pairs of vertices form a non-isolating (resp. isolating) cut. The dashed edge is irrelevant.
• Figure 2: An illustration of Case \ref{['case:gluingTwoC5']}. The solid blue edges are the matching $M$, while the dashed edge is the edge $w_1\hat{X}$. In the figure, one has that $X = C_2$.
• Figure 3: An illustration of Case \ref{['case:gluingOtherBlockC4']}. The red edges are removed from $S$. The red edge in $C_1$ is removed through the application of Corollary \ref{['cor:shortcutNonLocalC5']}.
• Figure 4: An illustration of Case \ref{['case:gluingOtherBlockOther']}. The red edge in $C_1$ is removed through the application of Corollary \ref{['cor:shortcutNonLocalC5']}.
• Figure 5: We illustrate how to compute a cycle of size at least $4$ that is incident to distinct nodes of $C_1$. In the figure, any of the dashed edges yields a cycle of the desired type.

Theorems & Definitions (78)

• Theorem 1
• Definition 1
• Lemma 1
• Lemma 2: 3-Matching Lemma GGJ23soda
• Definition 2: Canonical $2$-edge cover
• Lemma 3
• Lemma 4
• Lemma 5: Bridge Covering
• Lemma 6: Gluing
• proof : Proof of Theorem \ref{['thr:main']}