Maximization Problems Parameterized Using Their Minimization Versions: The Case of Vertex Cover

TL;DR

This paper explores a novel approach to analyzing maximization problems by parameterizing them using the solutions of their minimization counterparts, exemplified through the Maximal Minimal Vertex Cover problem and its weighted variant.

Contribution

It introduces a parameterized approximation algorithm for MMVC and its weighted variant, along with conditional lower bounds for their algorithmic running times.

Findings

Developed a parameterized approximation algorithm for MMVC.

Extended the approach to the weighted MMVC variant.

Provided conditional lower bounds for algorithmic complexity.

Abstract

The parameterized complexity of problems is often studied with respect to the size of their optimal solutions. However, for a maximization problem, the size of the optimal solution can be very large, rendering algorithms parameterized by it inefficient. Therefore, we suggest to study the parameterized complexity of maximization problems with respect to the size of the optimal solutions to their minimization versions. We examine this suggestion by considering the Maximal Minimal Vertex Cover (MMVC) problem, whose minimization version, Vertex Cover, is one of the most studied problems in the field of Parameterized Complexity. Our main contribution is a parameterized approximation algorithm for MMVC, including its weighted variant. We also give conditional lower bounds for the running times of algorithms for MMVC and its weighted variant.