  During recent years the field of fine-grained complexity has bloomed to
produce a plethora of results, with both applied and theoretical impact on the
computer science community. The cornerstone of the framework is the notion of
fine-grained reductions, which correlate the exact complexities of problems
such that improvements in their running times or hardness results are carried
over. We provide a parameterized viewpoint of these reductions (PFGR) in order
to further analyze the structure of improvable problems and set the foundations
of a unified methodology for extending algorithmic results. In this context, we
define a class of problems (FPI) that admit fixed-parameter improvements on
their running time. As an application of this framework we present a truly
sub-quadratic fixed-parameter algorithm for the orthogonal vectors problem.
Finally, we provide a circuit characterization for FPI to further solidify the
notion of improvement.
