A $2\ell k$ Kernel for $\ell$-Component Order Connectivity

TL;DR

The paper tackles the ell-Component Order Connectivity problem ($\ell$-COC) by developing a linear kernel of size at most $2\ell k$ for fixed $\ell$, computable in time $n^{O(\ell)}$; it further accelerates the kernel to $$(3e)^{\ell}\cdot n^{O(1)}$$ via a separation oracle. Central to the approach is a Weighted Expansion Lemma, a weighted generalization of the classic $q$-Expansion lemma, enabling the identification of reducible pairs $(X,Y)$ that can be safely condensed in the kernel. The authors connect LP relaxations to kernelization by showing that optimal LP solutions reveal reducible structures, and they introduce a polynomial-time algorithm to extract them; they also prove NP-hardness for a subproblem underpinning the separation oracle and provide deterministic derandomization of color coding to ensure practical, reproducible performance. Collectively, the work advances parameterized preprocessing for a natural graph vulnerability problem and offers novel techniques—weighted expansions, LP-guided reductions, and derandomized color coding—that may generalize to related kernelization tasks.

Abstract

In the $\ell$-Component Order Connectivity problem ($\ell \in \mathbb{N}$), we are given a graph $G$ on $n$ vertices, $m$ edges and a non-negative integer $k$ and asks whether there exists a set of vertices $S\subseteq V(G)$ such that $|S|\leq k$ and the size of the largest connected component in $G-S$ is at most $\ell$. In this paper, we give a linear programming based kernel for $\ell$-Component Order Connectivity with at most $2\ell k$ vertices that takes $n^{\mathcal{O}(\ell)}$ time for every constant $\ell$. Thereafter, we provide a separation oracle for the LP of $\ell$-COC implying that the kernel only takes $(3e)^{\ell}\cdot n^{O(1)}$ time. On the way to obtaining our kernel, we prove a generalization of the $q$-Expansion Lemma to weighted graphs. This generalization may be of independent interest.

Figures (1)

• Figure 1: Proof of Lemma \ref{['forrest']} and \ref{['stars']}. Cyclically shift smallest weight in a non-zero weight cycle to obtain a forest. Root each tree in the forest at a vertex in $A$ such that each vertex in $B$ has a parent in $A$. Assign the value of $v\in B$ to its parent $u\in A$. In this new assignment, a non-root vertex $u\in A$loses its parent $v_0\in B$ and $f(v_0u)\leq W-1$ which explains the cost of making a splitting assignment into an unsplitting assignment.

Theorems & Definitions (46)

• lemma 1
• proof
• lemma 2
• proof
• definition 1: $q$-expansion
• definition 2: Twin graph
• lemma 3
• lemma 4: Expansion Lemma pc_book
• lemma 5: folklore
• proof