Combining Crown Structures for Vulnerability Measures

TL;DR

The paper studies network vulnerability metrics VI, wVI, COC, and wCOC and develops crown-decomposition-based kernelization methods. By integrating multiple crown decompositions into an extended framework, it yields strong kernel bounds: a vertex kernel of size $3p^2$ for VI, $3(p^2 + p^{1.5} p_\ell)$ for wVI, and $3\mu(k + \sqrt{\mu}W)$ for wCOC, with $\mu=\max(k,W)$, plus a combinatorial $2kW$ kernel in FPT-time parameterized by the size of a maximum $(W+1)$-packing $r$. It further provides polynomial-time kernels for special cases: $W=1$ (vertex cover) and claw-free graphs, and a fully combinatorial approach that avoids LP-based methods. Overall, the work advances kernelization for graph vulnerability measures by leveraging enhanced crown-structure techniques and clarifying the role of packings in preprocessing reductions.

Abstract

Over the past decades, various metrics have emerged in graph theory to grasp the complex nature of network vulnerability. In this paper, we study two specific measures: (weighted) vertex integrity (wVI) and (weighted) component order connectivity (wCOC). These measures not only evaluate the number of vertices required to decompose a graph into fragments, but also take into account the size of the largest remaining component. The main focus of our paper is on kernelization algorithms tailored to both measures. We capitalize on the structural attributes inherent in different crown decompositions, strategically combining them to introduce novel kernelization algorithms that advance the current state of the field. In particular, we extend the scope of the balanced crown decomposition provided by Casel et al.~[7] and expand the applicability of crown decomposition techniques. In summary, we improve the vertex kernel of VI from $p^3$ to $p^2$, and of wVI from $p^3$ to $3(p^2 + p^{1.5} p_{\ell})$, where $p_{\ell} < p$ represents the weight of the heaviest component after removing a solution. For wCOC we improve the vertex kernel from $\mathcal{O}(k^2W + kW^2)$ to $3μ(k + \sqrtμW)$, where $μ= \max(k,W)$. We also give a combinatorial algorithm that provides a $2kW$ vertex kernel in FPT-runtime when parameterized by $r$, where $r \leq k$ is the size of a maximum $(W+1)$-packing. We further show that the algorithm computing the $2kW$ vertex kernel for COC can be transformed into a polynomial algorithm for two special cases, namely when $W=1$, which corresponds to the well-known vertex cover problem, and for claw-free graphs. In particular, we show a new way to obtain a $2k$ vertex kernel (or to obtain a 2-approximation) for the vertex cover problem by only using crown structures.

Figures (2)

• Figure 1: Left: Let $A = \{a_1,a_2,a_3,a_4\}$ be ordered in a top down manner, $w(Q) = 1$ for every $Q \in \mathbb{CC}(G[B])$ and $w(a) = 1$ for every $a \in A$. Then, $(\{a_1,a_2\},\{a_3,a_4\},1,f,\{3,1,3,4\})$ is a DBE, where the assignment $f$ are depicted with corresponding colored bold edges. Right: A $\lambda$-balanced crown decomposition, where the assignment $f$ are depicted with corresponding colored bold edges. The two dashed lines illustrate that $w(h) + w(f^{-1}(h)) > \lambda$ for every $h \in H$ while $w(Q) \leq \lambda$ for every $Q \in \mathbb{CC}(G[C])$.
• Figure 2: Embedding of the graph in the corresponding flow network with capacities, where $Q \in \mathbb{CC}(G[B])$.

Theorems & Definitions (77)

• Definition 1: Demanded balanced expansion and weighted crown decomposition
• Theorem 1: Demanded balanced expansion
• Definition 2: $\lambda$-balanced crown decomposition
• Theorem 2: Balanced crown decomposition theorem, DBLP:conf/esa/Casel0INZ21 Theorem 7
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 6
• proof
• Lemma 7