Monomials in arithmetic circuits: Complete problems in the counting hierarchy

TL;DR

This paper investigates the computational complexity of monomial detection and counting in polynomials represented by arithmetic circuits, establishing their completeness in certain subclasses of the counting hierarchy, especially for multilinear polynomials.

Contribution

It introduces natural complete problems for subclasses of the counting hierarchy related to monomial detection and counting in arithmetic circuits, including multilinear cases.

Findings

Monomial detection and counting are complete for specific counting hierarchy subclasses.

Results extend understanding of complexity in arithmetic circuit problems.

Provides new natural complete problems in the counting hierarchy.

Abstract

We consider the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of monomials. We show that these problems are complete for subclasses of the counting hierarchy which had few or no known natural complete problems. We also study these questions for circuits computing multilinear polynomials.