Matrix P-norms are NP-hard to approximate if p neq 1,2,infty

We show that for any rational p \in [1,\infty) except p = 1, 2, unless P = NP, there is no polynomial-time algorithm for approximating the matrix p-norm to arbitrary relative precision. We also show that for any rational p\in [1,\infty) including p = 1, 2, unless P = NP, there is no polynomial-time algorithm approximates the \infty, p mixed norm to some fixed relative precision.

💡 Research Summary

This paper investigates the computational hardness of approximating matrix norms that are defined by the ℓp‑vector norm, focusing on two families: the standard matrix p‑norm ‖A‖p = max_{‖x‖p=1}‖Ax‖p and the mixed ∞,p‑norm ‖A‖∞,p = max_i ‖A_{i*}‖p (the largest ℓp‑norm among the rows). The authors prove that, unless P = NP, no polynomial‑time algorithm can approximate these quantities to arbitrary relative accuracy when the exponent p is a rational number in the interval