The complexity of the fermionant, and immanants of constant width

TL;DR

This paper proves that computing the fermionant and certain immanants is computationally hard (#P-hard or P-hard), implying significant complexity barriers for these functions even on restricted graph classes.

Contribution

It establishes the computational hardness of the fermionant  and P-hardness of immanants of constant width, extending complexity results to these algebraic graph invariants.

Findings

Computing Ferm_k is #P-hard for k>2.

Computing Ferm_2 is P-hard.

Immanants of width 2 are hard to compute, with implications for complexity classes.

Abstract

In the context of statistical physics, Chandrasekharan and Wiese recently introduced the \emph{fermionant} $\Ferm_k$, a determinant-like quantity where each permutation $\pi$ is weighted by $-k$ raised to the number of cycles in $\pi$. We show that computing $\Ferm_k$ is #P-hard under Turing reductions for any constant $k > 2$, and is $\oplusP$-hard for $k=2$, even for the adjacency matrices of planar graphs. As a consequence, unless the polynomial hierarchy collapses, it is impossible to compute the immanant $\Imm_\lambda \,A$ as a function of the Young diagram $\lambda$ in polynomial time, even if the width of $\lambda$ is restricted to be at most 2. In particular, if $\Ferm_2$ is in P, or if $\Imm_\lambda$ is in P for all $\lambda$ of width 2, then $\NP \subseteq \RP$ and there are randomized polynomial-time algorithms for NP-complete problems.