# On the fixed-parameter tractability of the maximum connectivity improvement problem

## TL;DR

This paper investigates the parameterized complexity of the Maximum Connectivity Improvement problem on directed acyclic graphs, establishing hardness results and fixed-parameter tractability for certain parameters.

## Contribution

It provides the first fixed-parameter algorithms and hardness results for MCI on directed acyclic graphs based on various natural parameters.

## Key findings

- MCI is W[2]-hard for parameter B.
- MCI is fixed-parameter tractable for parameters |V|-B and ν.
- Characterization of MCI with respect to other parameters.

## Abstract

In the Maximum Connectivity Improvement (MCI) problem, we are given a directed graph $G=(V,E)$ and an integer $B$ and we are asked to find $B$ new edges to be added to $G$ in order to maximize the number of connected pairs of vertices in the resulting graph. The MCI problem has been studied from the approximation point of view. In this paper, we approach it from the parameterized complexity perspective in the case of directed acyclic graphs. We show several hardness and algorithmic results with respect to different natural parameters. Our main result is that the problem is $W[2]$-hard for parameter $B$ and it is FPT for parameters $|V| - B$ and $\nu$, the matching number of $G$. We further characterize the MCI problem with respect to other complementary parameters.