Approximating the cut-norm via Grothendieck's inequality

Noga Alon, Assaf Naor

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


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
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


OtherProceedings of the 36th Annual ACM Symposium on Theory of Computing
Country/TerritoryUnited States
CityChicago, IL


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

ASJC Scopus subject areas

  • Software


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