On Kernels for d-Path Vertex Cover

TL;DR

This paper improves kernelization bounds for the d-Path Vertex Cover problem, providing smaller kernels for specific cases and a more efficient general kernel, advancing parameterized complexity techniques.

Contribution

The paper presents new, smaller kernels for d-Path Vertex Cover, notably for d=4 and d=5, and a more efficient general kernel for arbitrary d.

Findings

Kernels with O(k^2) vertices and edges for d=4 and d=5.

A kernel with O(k^4 d^{2d+9}) vertices and edges for general d.

Enhanced kernelization bounds improve parameterized algorithms for d-Path Vertex Cover.

Abstract

In this paper we study the kernelization of the $d$-Path Vertex Cover ($d$-PVC) problem. Given a graph $G$, the problem requires finding whether there exists a set of at most $k$ vertices whose removal from $G$ results in a graph that does not contain a path (not necessarily induced) with $d$ vertices. It is known that $d$-PVC is NP-complete for $d\geq 2$. Since the problem generalizes to $d$-Hitting Set, it is known to admit a kernel with $\mathcal{O}(dk^d)$ edges. We improve on this by giving better kernels. Specifically, we give kernels with $\mathcal{O}(k^2)$ vertices and edges for the cases when $d=4$ and $d=5$. Further, we give a kernel with $\mathcal{O}(k^4d^{2d+9})$ vertices and edges for general $d$.