Parameterized Analogues of Probabilistic Computation

TL;DR

This paper introduces a new parameterized complexity class ${W[P]}$-${ m PFPT}$ as an analogue of ${ m PP}$, explores its properties, and applies it to polynomial identity testing and a parameterized permanent variant.

Contribution

It defines the class ${W[P]}$-${ m PFPT}$, translating properties of ${ m PP}$ to the parameterized setting, and studies related problems like polynomial identity testing and permanent variants.

Findings

Defined ${W[P]}$-${ m PFPT}$ as a parameterized analogue of ${ m PP}.

Established a parameterized Schwartz-Zippel lemma.

Proved the parameterized permanent variant is $ ext{ extsterling}W[1]$-complete.

Abstract

We study structural aspects of randomized parameterized computation. We introduce a new class ${\sf W[P]}$-${\sf PFPT}$ as a natural parameterized analogue of ${\sf PP}$. Our definition uses the machine based characterization of the parameterized complexity class ${\sf W[P]}$ obtained by Chen et.al [TCS 2005]. We translate most of the structural properties and characterizations of the class ${\sf PP}$ to the new class ${W[P]}$-${\sf PFPT}$. We study a parameterization of the polynomial identity testing problem based on the degree of the polynomial computed by the arithmetic circuit. We obtain a parameterized analogue of the well known Schwartz-Zippel lemma [Schwartz, JACM 80 and Zippel, EUROSAM 79]. Additionally, we introduce a parameterized variant of permanent, and prove its $\#W[1]$ completeness.