Approximating the cut-norm via Grothendieck's inequality

Noga Alon, Assaf Naor

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

Abstract

The cut-norm ∥A∥c of a real matrix A = (aij) iεR,jεS is the maximum, over all I ⊂ R, J ⊂ S of the quantity | ∑iεI, jεJ aij|. This concept plays a major role in the design of efficient approximation algorithms for dense graph and matrix problems. Here we show that the problem of approximating the cut-norm of a given real matrix is MAX SNP hard, and provide an efficient approximation algorithm. This algorithm finds, for a given matrix A = (a ij)iεR, jεs, two subsets I ⊂ R and J ⊂ S, such that | ∑iεI, jεJ aij| ≥ ρ ∥A∥c where ρ > 0 is an absolute constant satisfying ρ > 0.56. The algorithm combines semidefinite programming with a rounding technique based on Grothendieck's Inequality. We present three known proofs of Grothendieck's inequality, with the necessary modifications which emphasize their algorithmic aspects. These proofs contain rounding techniques which go beyond the random hyperplane rounding of Goemans and Williamson, allowing us to transfer various algorithms for dense graph and matrix problems to the sparse case.

Original languageEnglish (US)
Title of host publicationConference Proceedings of the Annual ACM Symposium on Theory of Computing
Pages72-80
Number of pages9
StatePublished - 2004
EventProceedings of the 36th Annual ACM Symposium on Theory of Computing - Chicago, IL, United States
Duration: Jun 13 2004Jun 15 2004

Other

OtherProceedings of the 36th Annual ACM Symposium on Theory of Computing
CountryUnited States
CityChicago, IL
Period6/13/046/15/04

Fingerprint

Approximation algorithms

Keywords

  • Cut-Norm
  • Grothendieck's Inequaity
  • Rounding Techniques

ASJC Scopus subject areas

  • Software

Cite this

Alon, N., & Naor, A. (2004). Approximating the cut-norm via Grothendieck's inequality. In Conference Proceedings of the Annual ACM Symposium on Theory of Computing (pp. 72-80)

Approximating the cut-norm via Grothendieck's inequality. / Alon, Noga; Naor, Assaf.

Conference Proceedings of the Annual ACM Symposium on Theory of Computing. 2004. p. 72-80.

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

Alon, N & Naor, A 2004, Approximating the cut-norm via Grothendieck's inequality. in Conference Proceedings of the Annual ACM Symposium on Theory of Computing. pp. 72-80, Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, United States, 6/13/04.
Alon N, Naor A. Approximating the cut-norm via Grothendieck's inequality. In Conference Proceedings of the Annual ACM Symposium on Theory of Computing. 2004. p. 72-80
Alon, Noga ; Naor, Assaf. / Approximating the cut-norm via Grothendieck's inequality. Conference Proceedings of the Annual ACM Symposium on Theory of Computing. 2004. pp. 72-80
@inproceedings{c46374812e2247d28d0186abc8d52913,
title = "Approximating the cut-norm via Grothendieck's inequality",
abstract = "The cut-norm ∥A∥c of a real matrix A = (aij) iεR,jεS is the maximum, over all I ⊂ R, J ⊂ S of the quantity | ∑iεI, jεJ aij|. This concept plays a major role in the design of efficient approximation algorithms for dense graph and matrix problems. Here we show that the problem of approximating the cut-norm of a given real matrix is MAX SNP hard, and provide an efficient approximation algorithm. This algorithm finds, for a given matrix A = (a ij)iεR, jεs, two subsets I ⊂ R and J ⊂ S, such that | ∑iεI, jεJ aij| ≥ ρ ∥A∥c where ρ > 0 is an absolute constant satisfying ρ > 0.56. The algorithm combines semidefinite programming with a rounding technique based on Grothendieck's Inequality. We present three known proofs of Grothendieck's inequality, with the necessary modifications which emphasize their algorithmic aspects. These proofs contain rounding techniques which go beyond the random hyperplane rounding of Goemans and Williamson, allowing us to transfer various algorithms for dense graph and matrix problems to the sparse case.",
keywords = "Cut-Norm, Grothendieck's Inequaity, Rounding Techniques",
author = "Noga Alon and Assaf Naor",
year = "2004",
language = "English (US)",
pages = "72--80",
booktitle = "Conference Proceedings of the Annual ACM Symposium on Theory of Computing",

}

TY - GEN

T1 - Approximating the cut-norm via Grothendieck's inequality

AU - Alon, Noga

AU - Naor, Assaf

PY - 2004

Y1 - 2004

N2 - The cut-norm ∥A∥c of a real matrix A = (aij) iεR,jεS is the maximum, over all I ⊂ R, J ⊂ S of the quantity | ∑iεI, jεJ aij|. This concept plays a major role in the design of efficient approximation algorithms for dense graph and matrix problems. Here we show that the problem of approximating the cut-norm of a given real matrix is MAX SNP hard, and provide an efficient approximation algorithm. This algorithm finds, for a given matrix A = (a ij)iεR, jεs, two subsets I ⊂ R and J ⊂ S, such that | ∑iεI, jεJ aij| ≥ ρ ∥A∥c where ρ > 0 is an absolute constant satisfying ρ > 0.56. The algorithm combines semidefinite programming with a rounding technique based on Grothendieck's Inequality. We present three known proofs of Grothendieck's inequality, with the necessary modifications which emphasize their algorithmic aspects. These proofs contain rounding techniques which go beyond the random hyperplane rounding of Goemans and Williamson, allowing us to transfer various algorithms for dense graph and matrix problems to the sparse case.

AB - The cut-norm ∥A∥c of a real matrix A = (aij) iεR,jεS is the maximum, over all I ⊂ R, J ⊂ S of the quantity | ∑iεI, jεJ aij|. This concept plays a major role in the design of efficient approximation algorithms for dense graph and matrix problems. Here we show that the problem of approximating the cut-norm of a given real matrix is MAX SNP hard, and provide an efficient approximation algorithm. This algorithm finds, for a given matrix A = (a ij)iεR, jεs, two subsets I ⊂ R and J ⊂ S, such that | ∑iεI, jεJ aij| ≥ ρ ∥A∥c where ρ > 0 is an absolute constant satisfying ρ > 0.56. The algorithm combines semidefinite programming with a rounding technique based on Grothendieck's Inequality. We present three known proofs of Grothendieck's inequality, with the necessary modifications which emphasize their algorithmic aspects. These proofs contain rounding techniques which go beyond the random hyperplane rounding of Goemans and Williamson, allowing us to transfer various algorithms for dense graph and matrix problems to the sparse case.

KW - Cut-Norm

KW - Grothendieck's Inequaity

KW - Rounding Techniques

UR - http://www.scopus.com/inward/record.url?scp=4544310407&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4544310407&partnerID=8YFLogxK

M3 - Conference contribution

SP - 72

EP - 80

BT - Conference Proceedings of the Annual ACM Symposium on Theory of Computing

ER -