Volume 9 (2013) Article 6 pp. 273-282 [Note]
The Complexity of the Fermionant and Immanants of Constant Width
Received: November 11, 2011
Revised: January 20, 2013
Published: February 26, 2013
In the context of statistical physics, Chandrasekharan and Wiese recently introduced the fermionant $\Ferm_k$, a determinant-like function of a matrix 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 polynomial-time Turing reductions for any constant $k > 2$, and is $\oplusP$-hard for $k=2$, where both results hold even for the adjacency matrices of planar graphs. As a consequence, unless the polynomial-time 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, unless $\NP \subseteq \RP$, $\Ferm_2$ is not in P, and there are Young diagrams $\lambda$ of width $2$ such that $\Imm_\lambda$ is not in P.