Matrix p -Norms Are NP-Hard to Approximate If $p eq1,2,\infty$
Julien M. Hendrickx, Alex Olshevsky · SIAM Journal on Matrix Analysis and Applications · 2010
We show that, for any rational $p\in[1,\infty)$ 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 $p\in[1,\infty)$ including $p=1,2$, unless $P=NP$, there is no polynomial-time algorithm which approximates the $\infty,p$ mixed norm to some fixed relative precision.