The Iris function and the matrix permanent

Abstract

This paper defines the Iris function and provides two formulations of the matrix permanent. The first formulation, valid for arbitrary complex matrices, expresses the permanent of a complex matrix as a contour integral of a second order Iris function over the unit circle around zero. The second formulation is defined for the restricted set of matrices with complex or "Gaussian" integer elements. Using the second formulation, the paper shows that the computation of the permanent of an arbitrary $n\times n$ 0-1 matrix is bounded by $o\left( n^{27} \left( \log \left( {{n^3}} \right) \right)^6 \left( \log_2 \left( {{n}} \right) \right)^2\right)$ binary operations.