# When can Graph Hyperbolicity be computed in Linear Time?

## TL;DR

This paper investigates the computational complexity of graph hyperbolicity, showing that it can be computed in linear time with fixed-parameter tractability, but not faster under certain complexity assumptions.

## Contribution

The paper introduces fixed-parameter algorithms for computing graph hyperbolicity in linear time based on graph parameters, advancing the understanding of its computational limits.

## Key findings

- Hyperbolicity can be computed in $O(2^{O(k)} + n + m)$ time.
- Unless SETH fails, no $2^{o(k)} n^2$-time algorithm exists.
- The results establish fixed-parameter tractability for hyperbolicity computation.

## Abstract

Hyperbolicity measures, in terms of (distance) metrics, how close a given graph is to being a tree. Due to its relevance in modeling real-world networks, hyperbolicity has seen intensive research over the last years. Unfortunately, the best known algorithms for computing the hyperbolicity number of a graph (the smaller, the more tree-like) have running time $O(n^4)$, where $n$ is the number of graph vertices. Exploiting the framework of parameterized complexity analysis, we explore possibilities for "linear-time FPT" algorithms to compute hyperbolicity. For instance, we show that hyperbolicity can be computed in time $O(2^{O(k)} + n +m)$ ($m$ being the number of graph edges) while at the same time, unless the SETH fails, there is no $2^{o(k)}n^2$-time algorithm.