On randomized counting versus randomised decision

TL;DR

This paper investigates the complexity of counting problems and their approximability via f.p.r.a.s., focusing on subclasses of #P with decision versions in P or RP, and explores the relationships between these classes.

Contribution

It characterizes the conditions under which certain counting problems admit f.p.r.a.s. based on their decision complexity classes and explores the structural relationships between these subclasses.

Findings

Identifies subclasses of #P with decision versions in P or RP.

Shows that #P problems with NP-complete decision versions do not admit f.p.r.a.s. unless NP=RP.

Analyzes the inclusion relations between classes like NP, RP, and P.

Abstract

We study the question of which counting problems admit f.p.r.a.s., under a structural complexity perspective. Since problems in #P with NP-complete decision version do not admit f.p.r.a.s. (unless NP = RP), we study subclasses of #P, having decision version either in P or in RP. We explore inclusions between these subclasses and we present all possible worlds with respect to NP v.s. RP and RP v.s. P.