Tight hardness of the non-commutative grothendieck problem

Jop Briët, Oded Regev, Rishi Saket

Research output: Contribution to journalArticle

Abstract

We prove that for any e > 0 it is NP-hard to approximate the non-commutative Grothendieck problem to within a factor 1=2+ε, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC’13). Our proof uses an embedding of ℓ2 into the space of matrices endowed with the trace norm with the property that the image of standard basis vectors is longer than that of unit vectors with no large coordinates. We also observe that one can obtain a tight NP-hardness result for the commutative Little Grothendieck problem; previously, this was only known based on the Unique Games Conjecture (Khot and Naor, Mathematika 2009).

Original languageEnglish (US)
JournalTheory of Computing
Volume13
DOIs
StatePublished - Dec 2 2017

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Hardness
Standard Basis
NP-hardness
Unit vector
NP-complete problem
Trace
Game
Norm
Approximation

Keywords

  • Grothendieck inequality
  • Hardness of approximation
  • Semidefinite programming

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics

Cite this

Tight hardness of the non-commutative grothendieck problem. / Briët, Jop; Regev, Oded; Saket, Rishi.

In: Theory of Computing, Vol. 13, 02.12.2017.

Research output: Contribution to journalArticle

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