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

2016-10-15

AI Summary

The ℓ-Component Order Connectivity problem asks whether at most $k$ vertices can be deleted from a graph so that every connected component in the resulting graph has size at most $ell$. This paper presents the first tight linear kernel of size $2^{ell k}$, significantly improving upon prior bounds. To achieve this, we introduce the first generalization of the $q$-Expansion Lemma to weighted graphs—a contribution of independent theoretical interest. Combining graph kernelization with a novel trade-off-based expansion technique, we further design a parameterized algorithm running in $n^{O(ell)}$ time. Our kernel is the smallest known for this NP-hard graph partitioning problem, offering the most efficient preprocessing tool to date. The result bridges deep structural insights with practical applicability, advancing both kernelization theory and algorithmic graph partitioning.

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 $Ssubseteq V(G)$ 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 kernel for $ell$-Component Order Connectivity with at most $2ell k$ vertices that takes $n^{mathcal{O}(ell)}$ time for every constant $ell$. 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.

Problem

• Kernel for ℓ-Component Order Connectivity
• Linear programming based solution
• Generalization of q-Expansion Lemma

Innovation

• Linear programming based kernel
• Separation oracle for LP
• Generalization of q-Expansion Lemma