MATRIX p-NORMS ARE NP-HARD TO APPROXIMATE IF p not equal to 1, 2, infinity

Abstract

We show that, for any rational p ∈ [1,∞) [p is an element of the set [1, infinity)] except p = 1, 2, unless P = NP, there is no polynomial time algorithm which approximates the matrix p-norm to arbitrary relative precision. We also show that, for any rational p ∈ [1,∞) [p is an element of the set [1, infinity)] including p = 1, 2, unless P = NP, there is no polynomialtime algorithm which approximates the ∞, p [infinity, p] mixed norm to some fixed relative precision.