#P-hardness proofs of matrix immanants evaluated on restricted matrices

Istvan Miklos, Cordian Riener

#P-hardness of computing matrix immanants are proved for each member of a broad class of shapes and restricted sets of matrices. The class is characterized in the following way. If a shape of size $n$ in it is in form $(w,\mathbf{1}+\lambda)$ or its conjugate is in that form, where $\mathbf{1}$ is the all-$1$ vector, then $|\lambda|$ is $n^{\varepsilon}$ for some $0<\varepsilon$, $\lambda$ can be tiled with $1\times 2$ dominos and $(3w+3h(\lambda)+1)|\lambda| \le n$, where $h(\lambda)$ is the height of $\lambda$. The problem remains \#P-hard if the immanants are evaluated on $0$-$1$ matrices. We also give hardness proofs of some immanants whose shape $\lambda = (\mathbf{1}+\lambda_d)$ has size $n$ such that $|\lambda_d| = n^{\varepsilon}$ for some $0<\varepsilon<\frac{1}{2}$, and for some $w$, the shape $\lambda_d/(w)$ is tilable with $1\times 2$ dominos. The \#P-hardness result holds when these immanants are evaluated on adjacency matrices of planar, directed graphs, however, in these cases the edges have small positive integer weights.

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