# An FPT Algorithm for Minimum Additive Spanner Problem

## TL;DR

This paper introduces a fixed-parameter algorithm for the NP-hard Minimum Additive t-Spanner Problem, focusing on the number of edges removed as a parameter, and extends results to $(eta, eta)$-spanners.

## Contribution

It provides the first fixed-parameter algorithm for the problem based on the number of edges removed, advancing the understanding of its parameterized complexity.

## Key findings

- Developed a fixed-parameter algorithm for the problem.
- Extended the algorithm to $(eta, eta)$-spanners.
- Contributed to the parameterized complexity analysis of additive spanners.

## Abstract

For a positive integer $t$ and a graph $G$, an additive $t$-spanner of $G$ is a spanning subgraph in which the distance between every pair of vertices is at most the original distance plus $t$. Minimum Additive $t$-Spanner Problem is to find an additive $t$-spanner with the minimum number of edges in a given graph, which is known to be NP-hard. Since we need to care about global properties of graphs when we deal with additive $t$-spanners, Minimum Additive $t$-Spanner Problem is hard to handle, and hence only few results are known for it. In this paper, we study Minimum Additive $t$-Spanner Problem from the viewpoint of parameterized complexity. We formulate a parameterized version of the problem in which the number of removed edges is regarded as a parameter, and give a fixed-parameter algorithm for it. We also extend our result to $(\alpha, \beta)$-spanners.