What is in #P and what is not?

TL;DR

This paper explores the boundaries of the counting complexity class #P, identifying cases of membership and non-membership, and studies algebraic closure properties and subclasses within #TFNP, advancing understanding of computational complexity.

Contribution

It establishes new results on #P membership and non-membership for classical functions, and initiates the study of algebraic closure properties of #P on affine varieties.

Findings

Proved #P membership for surprising cases

Established non-membership under standard assumptions

Demonstrated strict inclusion of #P in subclasses of #TFNP

Abstract

For several classical nonnegative integer functions, we investigate if they are members of the counting complexity class #P or not. We prove #P membership in surprising cases, and in other cases we prove non-membership, relying on standard complexity assumptions or on oracle separations. We initiate the study of the polynomial closure properties of #P on affine varieties, i.e., if all problem instances satisfy algebraic constraints. This is directly linked to classical combinatorial proofs of algebraic identities and inequalities. We investigate #TFNP and obtain oracle separations that prove the strict inclusion of #P in all standard syntactic subclasses of #TFNP-1.