# Parameterized Valiant's Classes

## TL;DR

This paper develops a theory of parameterized algebraic complexity classes, introduces new classes, and proves VW[1]-completeness of the parameterized clique family, involving complex proofs and new concepts.

## Contribution

It defines parameterized algebraic classes analogous to Boolean classes and establishes VW[1]-completeness for the parameterized clique family, with novel proof techniques.

## Key findings

- Defined classes VFPT and VW[t] analogous to #FPT and #W[t]
- Proved VW[1]-completeness of the parameterized clique family
- Introduced new concepts like composition theorems for bounded exponential sums

## Abstract

We define a theory of parameterized algebraic complexity classes in analogy to parameterized Boolean counting classes. We define the classes VFPT and VW[t], which mirror the Boolean counting classes #FPT and #W[t], and define appropriate reductions and completeness notions. Our main contribution is the VW[1]-completeness proof of the parameterized clique family. This proof is far more complicated than in the Boolean world. It requires some new concepts like composition theorems for bounded exponential sums and Boolean-arithmetic formulas. In addition, we also look at two polynomials linked to the permanent with vastly different parameterized complexity.