Determinant versus Permanent: Salvation via Generalization?

Abstract

The fermionant ${\rm Ferm}^k_n(\bar x) = \sum_{\sigma \in S_n} (-k)^{c(\pi)}\prod_{i=1}^n x_{i,j}$ can be seen as a generalization of both the permanent (for k = − 1) and the determinant (for k = 1). We demonstrate that it is $\textsc{VNP}$-complete for any rational k ≠ 1. Furthermore it is #P-complete for the same values of k. 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 ^ε boxes at the right of the first column is $\textsc{VNP}$-complete.