The complexity of the fermionant, and immanants of constant width

Details The complexity of the fermionant, and immanants of constant width The complexity of the fermionant, and immanants of constant width

2011-10-09

arxiv.org/abs/1110.1821v2

Computer Science

Computational Complexity

7 pages, 1 figure

Scientific paper

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.

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