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

TL;DR

This paper introduces a linear programming kernel of size at most 2ℓk for the ℓ-Component Order Connectivity problem, with efficient algorithms and a novel weighted graph expansion lemma.

Contribution

It presents a new kernelization technique for ℓ-Component Order Connectivity using LP, including a separation oracle and a generalized q-Expansion Lemma for weighted graphs.

Findings

Kernel size is at most 2ℓk vertices.

LP separation oracle reduces runtime to (3e)^ℓ·n^{O(1)}.

Generalized q-Expansion Lemma for weighted graphs of independent interest.

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.