# Faster FPT Algorithm for 5-Path Vertex Cover

## TL;DR

This paper introduces a faster fixed-parameter tractable algorithm for the 5-Path Vertex Cover problem, improving the running time from previous algorithms by employing an iterative compression technique.

## Contribution

The paper presents a novel iterative compression algorithm that reduces the running time for 5-PVC to D4^k n^{O(1)}, improving upon prior methods.

## Key findings

- Achieved D4^k n^{O(1)} running time for 5-PVC
- Improved fixed-parameter algorithm over previous approaches
- Demonstrated effectiveness of iterative compression for this problem

## Abstract

The problem of $d$-Path Vertex Cover, $d$-PVC lies in determining a subset $F$ of vertices of a given graph $G=(V,E)$ such that $G \setminus F$ does not contain a path on $d$ vertices. The paths we aim to cover need not to be induced. It is known that the $d$-PVC problem is NP-complete for any $d \ge 2$. When parameterized by the size of the solution $k$, 5-PVC has direct trivial algorithm with $\mathcal{O}(5^kn^{\mathcal{O}(1)})$ running time and, since $d$-PVC is a special case of $d$-Hitting Set, an algorithm running in $\mathcal{O}(4.0755^kn^{\mathcal{O}(1)})$ time is known. In this paper we present an iterative compression algorithm that solves the 5-PVC problem in $\mathcal{O}(4^kn^{\mathcal{O}(1)})$ time.