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Approximating the Cut-Norm via Grothendieck's Inequality Assaf Naor
 

Summary: Approximating the Cut-Norm via Grothendieck's Inequality
Noga Alon
Assaf Naor
Abstract
The cut-norm ||A||C of a real matrix A = (aij)iR,jS is the maximum, over all I R, J
S of the quantity | iI,jJ 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 = (aij)iR,jS, two subsets
I R and J S, such that | iI,jJ aij| ||A||C, where > 0 is an absolute constant
satisfying > 0.56. The algorithm combines semidefinite programming with a novel rounding
technique based on Grothendieck's Inequality.
1 Introduction
The cut-norm ||A||C of a real matrix A = (aij)iR,jS with a set of rows indexed by R and a set of
columns indexed by S is the maximum, over all I R, J S, of the quantity | iI,jJ aij|. This
concept plays a major role in the work of Frieze and Kannan on efficient approximation algorithms
for dense graph and matrix problems, [7] (see also [2] and its references). Although the techniques
in [7] enable the authors to approximate efficiently the cut-norm of an n by m matrix with entries in
[-1, 1] up to an additive error of nm, there is no known polynomial algorithm that approximates
the cut-norm of a general real matrix up to a constant multiplicative factor.

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics