Counting Problems Computationally Equivalent to Computing the Determinant

The main purpose of this paper is to exhibit non-algebraic problems that are computationally equivalent to computing the integer determinant. For this purpose, some graph-theoretic counting problems are shown to be equivalent to the integer determinant problem under suitable reducibilities. Those are the problems of counting the number of all paths between two nodes of a given acyclic digraph, the number of all smallest length paths between two nodes of a given undirected graph, the number of rooted spanning trees of a given digraph, and the number of Eulerian paths in a given digraph. It is also observed that the integer determinant problem (and some well-known linear algebraic problems) remains its essential complexity even if we require all entries of a given matrix to be either zero or one.