Determinant versus Permanent: salvation via generalization? The algebraic complexity of the Fermionant and the Immanant

TL;DR

This paper investigates the algebraic complexity of the fermionant and immanant, showing their computational hardness and generalization of the permanent and determinant, with implications for complexity theory.

Contribution

It establishes the VNP-completeness of the fermionant and immanant in various cases, advancing understanding of their computational complexity.

Findings

Fermionant is VNP-complete for most cases.

Fermionant is #P-complete for certain cases.

Immanant with bounded width Young diagrams is VNP-complete.

Abstract

The fermionant can be seen as a generalization of both the permanent (for $k=-1$) and the determinant. We demonstrate that it is VNP-complete for most cases. Furthermore it is #P-complete for the cases. The immanant is also a generalization of the permanent (for a Young diagram with a single line) and of the determinant (when the Young diagram is a column). We demonstrate that the immanant of any family of Young diagrams with bounded width and at least n boxes at the right of the first column is VNP-complete.