### 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 ℓ_{2} 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 language | English (US) |
---|---|

Journal | Theory of Computing |

Volume | 13 |

DOIs | |

State | Published - Dec 2 2017 |

### Fingerprint

### Keywords

- Grothendieck inequality
- Hardness of approximation
- Semidefinite programming

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Theory of Computing*,

*13*. https://doi.org/10.4086/toc.2017.v013a015

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

Research output: Contribution to journal › Article

*Theory of Computing*, vol. 13. https://doi.org/10.4086/toc.2017.v013a015

}

TY - JOUR

T1 - Tight hardness of the non-commutative grothendieck problem

AU - Briët, Jop

AU - Regev, Oded

AU - Saket, Rishi

PY - 2017/12/2

Y1 - 2017/12/2

N2 - 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).

AB - 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).

KW - Grothendieck inequality

KW - Hardness of approximation

KW - Semidefinite programming

UR - http://www.scopus.com/inward/record.url?scp=85044717906&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85044717906&partnerID=8YFLogxK

U2 - 10.4086/toc.2017.v013a015

DO - 10.4086/toc.2017.v013a015

M3 - Article

VL - 13

JO - Theory of Computing

JF - Theory of Computing

SN - 1557-2862

ER -