Computation of Matrix Permanent with Automatic Differentiation

Abstract

With a well-known formulation of matrix permanent by a multivariate polynomial, algorithms for the computation of the matrix permanent are considered in terms of automatic differentiation, where a succinct program with a C++ template for the higher order derivatives is described. A special set of commutative quadratic nilpotent elements is introduced, and it is shown that the permanent can be computed efficiently as a variation of implementation of higher order automatic differentiation. Given several ways for transforming the multivariate polynomial into univariate polynomials, six algorithms that compute the value of the permanent are described with their computational complexities. One of the complexities is O(n2n), the same as that of the most popular Ryser's algorithm.