On commutativity and approximation

Abstract

The action of commutativity and approximation is analyzed for some problems in Computational Complexity. Lower bound criteria to the approximate complexity are given in terms of border rank and commulative border rank of a given tensor. Upper bounds for the approximate complexity of the matrix-vector product are given. In particular, 1 2 m(n+1) multiplications are necessary and sufficient to approximate n × m matrix-vector product; 6 multiplications are sufficient (5 are needed) to approximate a 2 × 2 matrix product by using commutativity. An application to polynomial evaluation shows that 1 2 n+2 multiplications are sufficient to approximate any n-degree polynomial at a point. For what concerns matrix multiplication complexity a number θ is introduced such that θ⩽ ω ( ω is the exponent of matrix multiplication complexity). This number measures the degree of complexity of the best commutative approximate algorithm for matrix multiplication. The bound θ⩽2.3211…and conditions under which θ= ω are shown.