A note on the hardness of graph diameter augmentation problems

TL;DR

This paper demonstrates that the problem of augmenting a graph's edges to achieve a specific diameter is computationally hard, providing a simpler proof of NP-hardness and establishing W[2]-hardness.

Contribution

The paper offers a simplified reduction for the NP-hardness of the Diameter-D Augmentation problem and proves its W[2]-hardness, advancing understanding of its computational complexity.

Findings

Diameter-D Augmentation is NP-hard even for D=2

The problem is W[2]-hard in parameterized complexity

Simplified reduction technique for hardness proof

Abstract

A graph has \emph{diameter} D if every pair of vertices are connected by a path of at most D edges. The Diameter-D Augmentation problem asks how to add the a number of edges to a graph in order to make the resulting graph have diameter D. It was previously known that this problem is NP-hard \cite{GJ}, even in the D=2 case. In this note, we give a simpler reduction to arrive at this fact and show that this problem is W[2]-hard.