Determinant: Combinatorics, Algorithms, and Complexity

Abstract

We prove a new combinatorial characterization of the determinant. The characterization yields a simple combinatorial algorithm for computing the determinant. Hitherto, all (known) algorithms for the determinant have been based on linear algebra. Our combinatorial algorithm requires no division, and works over arbitrary commutative rings. It also lends itself to efficient parallel implementations. It has been known for some time now that the complexity class GapL characterizes the complexity of computing the determinant of matrices over the integers. We present a direct proof of this characterization.