# Parameterized Fine-Grained Reductions

## TL;DR

This paper introduces a parameterized framework for fine-grained reductions, enabling a deeper analysis of problem structures and fixed-parameter improvements, exemplified by a sub-quadratic algorithm for the orthogonal vectors problem.

## Contribution

It proposes a unified parameterized approach to fine-grained reductions, defines the class FPI for problems with fixed-parameter improvements, and provides a circuit characterization for FPI.

## Key findings

- Developed a parameterized framework for fine-grained reductions.
- Established FPI as a class of problems with fixed-parameter improvements.
- Presented a sub-quadratic fixed-parameter algorithm for the orthogonal vectors problem.

## Abstract

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.