# Solving Vertex Cover in Polynomial Time on Hyperbolic Random Graphs

## TL;DR

This paper demonstrates that the Vertex Cover problem can be solved in polynomial time on hyperbolic random graphs, providing insights into real-world network structures and improving approximation algorithms.

## Contribution

It proves polynomial-time solvability of Vertex Cover on hyperbolic random graphs and links structural properties to practical algorithm performance.

## Key findings

- Vertex Cover solvable in polynomial time on hyperbolic random graphs
- Structural properties observed in real-world networks
- Adaptive greedy algorithms outperform standard approaches

## Abstract

The VertexCover problem is proven to be computationally hard in different ways: It is NP-complete to find an optimal solution and even NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on many real-world networks the run time to solve VertexCover is way smaller than even the best known FPT-approaches can explain. Similarly, greedy algorithms deliver very good approximations to the optimal solution in practice. We link these observations to two properties that are observed in many real-world networks, namely a heterogeneous degree distribution and high clustering. To formalize these properties and explain the observed behavior, we analyze how a branch-and-reduce algorithm performs on hyperbolic random graphs, which have become increasingly popular for modeling real-world networks. In fact, we are able to show that the VertexCover problem on hyperbolic random graphs can be solved in polynomial time, with high probability. The proof relies on interesting structural properties of hyperbolic random graphs. Since these predictions of the model are interesting in their own right, we conducted experiments on real-world networks showing that these properties are also observed in practice. When utilizing the same structural properties in an adaptive greedy algorithm, further experiments suggest that, on real instances, this leads to better approximations than the standard greedy approach within reasonable time.