# Rank Vertex Cover as a Natural Problem for Algebraic Compression

## TL;DR

This paper introduces an algebraic compression approach for the Vertex Cover Above LP problem, transforming it into the Rank Vertex Cover problem using matroid representations, offering a new perspective in parameterized complexity.

## Contribution

It presents an alternative algebraic compression of Vertex Cover Above LP into Rank Vertex Cover, connecting graph problems with matroid theory.

## Key findings

- Provides an algebraic compression method for Vertex Cover Above LP
- Establishes a connection between vertex cover problems and matroid rank functions
- Offers a new framework for parameterized complexity analysis

## Abstract

The question of the existence of a polynomial kernelization of the Vertex Cover Above LP problem has been a longstanding, notorious open problem in Parameterized Complexity. Five years ago, the breakthrough work by Kratsch and Wahlstrom on representative sets has finally answered this question in the affirmative [FOCS 2012]. In this paper, we present an alternative, algebraic compression of the Vertex Cover Above LP problem into the Rank Vertex Cover problem. Here, the input consists of a graph G, a parameter k, and a bijection between V (G) and the set of columns of a representation of a matriod M, and the objective is to find a vertex cover whose rank is upper bounded by k.