# Integrality Gap of the Vertex Cover Linear Programming Relaxation

## TL;DR

This paper characterizes the integrality gap of the vertex cover LP relaxation, linking it to the fractional chromatic number of the graph, providing a precise mathematical relationship.

## Contribution

It establishes a exact formula for the integrality gap of the vertex cover LP relaxation based on the fractional chromatic number, a novel theoretical insight.

## Key findings

- Integrality gap equals 2 - 2/χ^f(G) for any graph G.
- Connects the integrality gap to fractional chromatic number.
- Provides a precise characterization of LP relaxation performance.

## Abstract

We give a characterization result for the integrality gap of the natural linear programming relaxation for the vertex cover problem. We show that integrality gap of the standard linear programming relaxation for any graph G equals $\left(2-\frac{2}{\chi^f(G)}\right)$ where $\chi^f(G)$ denotes the fractional chromatic number of G.