Algebra and Complexity

Abstract

In this survey we shall discuss the relation between algebra and complexity by looking at a ubiquitous computational problem: Matrix multiplication. Our point of view will be asymptotic with regard to the size of the matrices considered. For simplicity we will work over the field of complex numbers, although the main results covered in this paper hold over fields of any characteristic, algebraically closed or not. Once the ground field is fixed we may define the so-called exponent ω of matrix multiplication, which controls the computational complexity of multiplying large matrices:

$$ \omega : = \inf \left\{ {\tau :L\left( {\left\langle {h,h,h} \right\rangle } \right) = O\left( {{h^\tau }} \right)} \right\} $$

(1)

Here 〈h,h,h〉 stands for the multiplication map of matrices of order h and L(〈h,h,h〉) denotes its complexity, i.e., the minimal number of arithmetical operations sufficient to compute the product of two generic matrices (by straight line algorithms). Thus ω is the smallest number  τ, “smallest” in the sense of infimum, such that matrices of sufficiently high order h may be multiplied by an algorithm using only h ^τ arithmetic operations.