### Abstract

The cut-norm ∥A∥c of a real matrix A = (a_{ij}) _{iεR,jεS} is the maximum, over all I ⊂ R, J ⊂ S of the quantity | ∑_{iεI, jεJ} a_{ij}|. 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} a_{ij}| ≥ 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 language | English (US) |
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Title of host publication | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |

Pages | 72-80 |

Number of pages | 9 |

State | Published - 2004 |

Event | Proceedings of the 36th Annual ACM Symposium on Theory of Computing - Chicago, IL, United States Duration: Jun 13 2004 → Jun 15 2004 |

### Other

Other | Proceedings of the 36th Annual ACM Symposium on Theory of Computing |
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Country | United States |

City | Chicago, IL |

Period | 6/13/04 → 6/15/04 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Software

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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.

}

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 -