Division-Free Algorithms for the Determinant and the Pfaffian: Algebraic and Combinatorial Approaches

shows that the determinant can be computed without divisions. The summation is taken over the set of all permutations π of n elements. Avoiding divisions seems attractive when working over a commutative ring which is not a field, for example when the entries are integers, polynomials, or rational or even more complicated expressions. Such determinants arise in combinatorial problems, see [11]. We will describe an O(n4) algorithm that works without divisions, and we will look at this algorithm from a combinatorial and an algebraic viewpoint. We will also consider the Pfaffian of a skew-symmetric matrix, a quantity closely related to the determinant. The results are in many ways analogous to those for the determinant, but the algebraic aspect of the algorithms is not explored yet. This survey is yet another article which highlights the close connection between linear algebra and cycles, paths, and matchings in graphs [1,17,25]. Much of this material is based on the papers of Mahajan and Vinay [14,15] and of Mahajan, Subramanya, and Vinay [13].