Improved Approximations for Flexible Network Design

TL;DR

The authors address flexible connectivity problems where a graph is partitioned into safe and unsafe elements, focusing on both edge-connectivity (FGC) and vertex-connectivity (FVC) variants and their higher-connectivity generalizations. They introduce two approximation schemes for FVC, achieving a first factor strictly below 2 via APX1 ($\approx$ $5/3$, with a $4/3$-case) and APX2 ($11/7$) that leverages ear-decompositions and rainbow-connection techniques. For FGC, they refine existing analyses to obtain a tighter bound (notably $10/7$) via a refined decomposition of safe and unsafe parts and contraction-based arguments, and they extend to a $1+O\left(\frac{1}{\sqrt{k}}\right)$-approximation for $k$-FGC. The work combines ear-decomposition, block theory, matroid intersection, and rainbow-connection concepts to push closer to optimal resilience with scalable guarantees, impacting design of fault-tolerant networks under partial failure models.

Abstract

Flexible network design deals with building a network that guarantees some connectivity requirements between its vertices, even when some of its elements (like vertices or edges) fail. In particular, the set of edges (resp. vertices) of a given graph are here partitioned into safe and unsafe. The goal is to identify a minimum size subgraph that is 2-edge-connected (resp. 2-vertex-connected), and stay so whenever any of the unsafe elements gets removed. In this paper, we provide improved approximation algorithms for flexible network design problems, considering both edge-connectivity and vertex-connectivity, as well as connectivity values higher than 2. For the vertex-connectivity variant, in particular, our algorithm is the first with approximation factor strictly better than 2.

Figures (3)

• Figure 2: A depiction of the edges and vertex sets found by Algorithms \ref{['alg:phase1']}, \ref{['alg:phase2']}, and \ref{['alg:phase3']} in $V(D)$. Here the unsafe vertices are depicted by black circles. In this example there is only one safe vertex, $v_1$ in the set $V(D)$ that is shown by a square. (a) The dashed edges are pseudo-edges $P$ found by Lemma \ref{['lem:RainbowForest']}. Algorithm \ref{['alg:phase1']} first computes good cycle on green edges that merges two large components of pseudo-edges, then it finds the red cycle that merges the new large component and 2 singletons. $X_1 = \{u_1, v_1\}$ (b) The yellow edges of the second figure are found by Algorithm \ref{['alg:phase2']} which cover the cut-vertex $c$ in the component. The interior vertex is $X_2 = \{u_2\}$. (c) The blue edges of the third figure are the edges found by Algorithm \ref{['alg:phase3']}, which add edges to the solution that bring $u_3$ and $v_3$ into $V(D)$ form a feasible FVC solution. The vertex $v_1$ is a safe vertex so we only add one edge ($x_3v_1$) incident on $v_3$.
• Figure 4: As in the statement and proof of Lemma \ref{['lem:forbiddencyclesUnsafe']}, we are given a forbidden cycle $C$, with unsafe vertices $u$ and $v$, and degree 2 vertices $w$ and $z$. The edges of $F'$ are shown with solid edges, and $F$ is both the solid lines and the dashed edges incident to $w$. Since $G\geq 5$, there is an additional vertex $x\notin V(C)$. We wish to show that $F'$ neither $u$ nor $z$ is a cut vertex of $(V\setminus\{w\}, F')$ showing that if so, then that vertex will be cut vertex separating $v$ from $x$ in . We consider cases if $u$ or $z$ are cut vertices of $(V\setminus\{w\}, F')$: (a) $u$ is a cut vertex (shown as a square), separating $x$ from $z$ and $v$, and clearly even with $vw$, $x$ is still separated from these vertices. (b) $z$ is a cut vertex(shown as a square), separating $v$ from $u$, and in particular, separating $v$ from $x$. It is clear that in this case $u$ is again a cut vertex.
• Figure 5: As in the statement and proof of Lemma \ref{['lem:forbiddencycles']}, we are given a forbidden cycle $C$, with unsafe vertices $u$ and $v$, and degree 2 vertices $w$ and $z$. (a) in the first figure, the solid edges represent edges in $F$, and the dashed represent the edges not in $F$ but in $C$. In the second figure we have the solid edges representing $F' \coloneqq F\backslash\{uw\} \cup \{vw\}$. Since $v$ is safe, we can replace $uw$ with $vw$, and not create an unsafe cut vertex. So $F'$ is a feasible FVC solution. (b) Here the solid edges again represent edges of $F'$. The green edges show a path from $u$ to $v$ that does not contain an edge of $C$. Here we depict the case that $z$ is an unsafe cut vertex of $F'$. Since $G$ is $2$VC, we can pick edge $e$ of this path that connects the two components of $F'\backslash\{z\}$.

Theorems & Definitions (44)

• Definition 1: Flexible Graph Connectivity Problem (FGC)
• Definition 2: Flexible Vertex Connectivity Problem (FVC)
• Theorem 3
• Theorem 4
• Definition 5: $k$-Flexible Graph Connectivity Problem ($k$-FGC)
• Theorem 6
• Definition 7: Ear-Decomposition
• Lemma 8: west2001introduction
• Definition 9: Blocks
• Lemma 10: west2001introduction