Approximating the cut-norm via Grothendieck's inequality

Noga Alon, Assaf Naor
2004 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing - STOC '04  
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 | 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 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
more » ... at | i∈I,j∈J a ij | ≥ ρ||A|| C , where ρ > 0 is an absolute constant satisfying ρ > 0.56. The algorithm combines semidefinite programming with a novel rounding technique based on Grothendieck's Inequality.
doi:10.1145/1007352.1007371 dblp:conf/stoc/AlonN04 fatcat:ndhbtsbzangl3dri366xthopti