Tight Hardness of the Non-commutative Grothendieck Problem

Jop Briot, Oded Regev, Rishi Saket

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We prove that it is NP-hard to approximate the non-commutative Grothendieck problem to within any constant factor larger than one-half, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of finite-dimensional Hilbert spaces 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.

Original languageEnglish (US)
Title of host publicationProceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015
PublisherIEEE Computer Society
Pages1108-1122
Number of pages15
Volume2015-December
ISBN (Print)9781467381918
DOIs
StatePublished - Dec 11 2015
Event56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 - Berkeley, United States
Duration: Oct 17 2015Oct 20 2015

Other

Other56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015
CountryUnited States
CityBerkeley
Period10/17/1510/20/15

Fingerprint

Hardness
Hilbert spaces

Keywords

  • Grothendieck inequality
  • hardness
  • semidefinite programming

ASJC Scopus subject areas

  • Computer Science(all)

Cite this

Briot, J., Regev, O., & Saket, R. (2015). Tight Hardness of the Non-commutative Grothendieck Problem. In Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015 (Vol. 2015-December, pp. 1108-1122). [7354446] IEEE Computer Society. https://doi.org/10.1109/FOCS.2015.72

Tight Hardness of the Non-commutative Grothendieck Problem. / Briot, Jop; Regev, Oded; Saket, Rishi.

Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015. Vol. 2015-December IEEE Computer Society, 2015. p. 1108-1122 7354446.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Briot, J, Regev, O & Saket, R 2015, Tight Hardness of the Non-commutative Grothendieck Problem. in Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015. vol. 2015-December, 7354446, IEEE Computer Society, pp. 1108-1122, 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, United States, 10/17/15. https://doi.org/10.1109/FOCS.2015.72
Briot J, Regev O, Saket R. Tight Hardness of the Non-commutative Grothendieck Problem. In Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015. Vol. 2015-December. IEEE Computer Society. 2015. p. 1108-1122. 7354446 https://doi.org/10.1109/FOCS.2015.72
Briot, Jop ; Regev, Oded ; Saket, Rishi. / Tight Hardness of the Non-commutative Grothendieck Problem. Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015. Vol. 2015-December IEEE Computer Society, 2015. pp. 1108-1122
@inproceedings{d6a5a08e9f3c4c30882f9cecb5c607c5,
title = "Tight Hardness of the Non-commutative Grothendieck Problem",
abstract = "We prove that it is NP-hard to approximate the non-commutative Grothendieck problem to within any constant factor larger than one-half, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of finite-dimensional Hilbert spaces 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.",
keywords = "Grothendieck inequality, hardness, semidefinite programming",
author = "Jop Briot and Oded Regev and Rishi Saket",
year = "2015",
month = "12",
day = "11",
doi = "10.1109/FOCS.2015.72",
language = "English (US)",
isbn = "9781467381918",
volume = "2015-December",
pages = "1108--1122",
booktitle = "Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015",
publisher = "IEEE Computer Society",

}

TY - GEN

T1 - Tight Hardness of the Non-commutative Grothendieck Problem

AU - Briot, Jop

AU - Regev, Oded

AU - Saket, Rishi

PY - 2015/12/11

Y1 - 2015/12/11

N2 - We prove that it is NP-hard to approximate the non-commutative Grothendieck problem to within any constant factor larger than one-half, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of finite-dimensional Hilbert spaces 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.

AB - We prove that it is NP-hard to approximate the non-commutative Grothendieck problem to within any constant factor larger than one-half, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of finite-dimensional Hilbert spaces 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.

KW - Grothendieck inequality

KW - hardness

KW - semidefinite programming

UR - http://www.scopus.com/inward/record.url?scp=84960423518&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84960423518&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2015.72

DO - 10.1109/FOCS.2015.72

M3 - Conference contribution

SN - 9781467381918

VL - 2015-December

SP - 1108

EP - 1122

BT - Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015

PB - IEEE Computer Society

ER -