Kernelization for Partial Vertex Cover via (Additive) Expansion Lemma

TL;DR

This paper introduces kernelization techniques for the Partial Vertex Cover problem using the expansion lemma and its additive variant, resulting in smaller problem kernels with fewer vertices.

Contribution

It presents a new additive expansion lemma and applies it to improve kernel size bounds for Partial Vertex Cover.

Findings

Kernel with $(\ell + 2)(k + \ell)$ vertices using the expansion lemma.

Smaller kernel with $(\ell + 1)(k + \ell)$ vertices via the additive expansion lemma.

The additive expansion lemma enhances kernelization bounds for Partial Vertex Cover.

Abstract

Given a graph and two integers $k$ and $\ell$, Partial Vertex Cover asks for a set of at most $k$ vertices whose deletion results in a graph with at most $\ell$ edges. Based on the expansion lemma, we provide a problem kernel with $(\ell + 2)(k + \ell)$ vertices. We then introduce a new, additive version of the expansion lemma and show it can be used to prove a kernel with $(\ell + 1)(k + \ell)$ vertices for $\ell \ge 1$.