  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.
