Matrix $p$-Norms Are NP-Hard to Approximate If $p\neq1,2,\infty$

  • Authors:

  • Julien M. Hendrickx;Alex Olshevsky

  • Affiliations:

  • julien.hendrickx@uclouvain.be;aolshevs@princeton.edu

  • Venue:
  • SIAM Journal on Matrix Analysis and Applications
  • Year:
  • 2010

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Abstract

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.