### Abstract

The cut-norm ∥A∥
_{c} of a real matrix A = (a
_{ij})
_{i∈R,j∈S} is the maximum, over all I ⊂ R, J ⊂ S, of the quantity | Σ
_{i∈I, j∈J}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 we provide an efficient approximation algorithm. This algorithm finds, for a given matrix A = (a
_{ij})
_{i∈R,j∈S}, two subsets I ⊂ R and J ⊂ S, such that | Σ
_{i∈I,j∈J} a
_{ij}| ≥ρ∥
_{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 [J. ACM, 42 (1995), pp. 1115-1145], allowing us to transfer various algorithms for dense graph and matrix problems to the sparse case.

Original language | English (US) |
---|---|

Pages (from-to) | 787-803 |

Number of pages | 17 |

Journal | SIAM Journal on Computing |

Volume | 35 |

Issue number | 4 |

DOIs | |

State | Published - 2006 |

### Fingerprint

### Keywords

- Approximation algorithms
- Cut-norm
- Grothendieck's inequality
- Semidefinite programming
- Szemerédi partitions

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Computing*,

*35*(4), 787-803. https://doi.org/10.1137/S0097539704441629

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

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 35, no. 4, pp. 787-803. https://doi.org/10.1137/S0097539704441629

}

TY - JOUR

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

AU - Alon, Noga

AU - Naor, Assaf

PY - 2006

Y1 - 2006

N2 - The cut-norm ∥A∥ c of a real matrix A = (a ij) i∈R,j∈S is the maximum, over all I ⊂ R, J ⊂ S, of the quantity | Σ i∈I, j∈Ja 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 we provide an efficient approximation algorithm. This algorithm finds, for a given matrix A = (a ij) i∈R,j∈S, two subsets I ⊂ R and J ⊂ S, such that | Σ i∈I,j∈J a ij| ≥ρ∥ 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 [J. ACM, 42 (1995), pp. 1115-1145], 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 = (a ij) i∈R,j∈S is the maximum, over all I ⊂ R, J ⊂ S, of the quantity | Σ i∈I, j∈Ja 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 we provide an efficient approximation algorithm. This algorithm finds, for a given matrix A = (a ij) i∈R,j∈S, two subsets I ⊂ R and J ⊂ S, such that | Σ i∈I,j∈J a ij| ≥ρ∥ 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 [J. ACM, 42 (1995), pp. 1115-1145], allowing us to transfer various algorithms for dense graph and matrix problems to the sparse case.

KW - Approximation algorithms

KW - Cut-norm

KW - Grothendieck's inequality

KW - Semidefinite programming

KW - Szemerédi partitions

UR - http://www.scopus.com/inward/record.url?scp=33747188137&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33747188137&partnerID=8YFLogxK

U2 - 10.1137/S0097539704441629

DO - 10.1137/S0097539704441629

M3 - Article

VL - 35

SP - 787

EP - 803

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 4

ER -