Approximating the cut-norm via Grothendieck's inequality

The <i>cut-norm</i> ||A||<inf>C</inf> of a real matrix A=(a<inf>ij</inf>)<inf>i∈ R,j∈S</inf> is the maximum, over all I ⊂ R, J ⊂ S of the quantity | Σ<inf>i ∈ I, j ∈ J</inf> a<inf>ij</inf>|. 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<inf>ij</inf>)<inf>i ∈ R, j ∈ S</inf>, two subsets I ⊂ R and J ⊂ S, such that | Σ<inf>i ∈ I, j ∈ J</inf> a<inf>ij</inf>| ≥ ρ ||A||<inf>C</inf>, 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 [12], allowing us to transfer various algorithms for dense graph and matrix problems to the sparse case.

[1]  J. Lindenstrauss,et al.  Absolutely summing operators in Lp spaces and their applications , 1968 .

[2]  R. Rietz.  A proof of the Grothendieck inequality , 1974 .

[3]  E. Szemerédi.  Regular Partitions of Graphs , 1975 .

[4]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[5]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[6]  U. Haagerup.  A new upper bound for the complex Grothendieck constant , 1987 .

[7]  G. Jameson.  Summing and nuclear norms in Banach space theory , 1987 .

[8]  H. König.  Geometry of Banach Spaces: On the complex Grothendieck constant in the n-dimensional case , 1991 .

[9]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[10]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[11]  Vojtech Rödl,et al.  The Algorithmic Aspects of the Regularity Lemma , 1994, J. Algorithms.

[12]  J. Diestel,et al.  Absolutely Summing Operators , 1995 .

[13]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[14]  A. Grothendieck.  Résumé de la théorie métrique des produits tensoriels topologiques , 1996 .

[15]  Y. Nesterov.  Semidefinite relaxation and nonconvex quadratic optimization , 1998 .

[16]  Noga Alon,et al.  The Space Complexity of Approximating the Frequency Moments , 1999 .

[17]  Alan M. Frieze,et al.  Quick Approximation to Matrices and Applications , 1999, Comb..

[18]  Sanjeev Mahajan,et al.  Derandomizing Approximation Algorithms Based on Semidefinite Programming , 1999, SIAM J. Comput..

[19]  Uri Zwick,et al.  Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems , 1999, STOC '99.

[20]  N. Alon,et al.  The Probabilistic Method, Second Edition , 2000 .

[21]  J. Lindenstrauss,et al.  Basic Concepts in the Geometry of Banach Spaces , 2001 .

[22]  Michael Langberg,et al.  The RPR2 rounding technique for semidefinite programs , 2001, J. Algorithms.

[23]  Johan Håstad,et al.  Some optimal inapproximability results , 2001, JACM.

[24]  Noga Alon,et al.  Random sampling and approximation of MAX-CSP problems , 2002, STOC '02.

[25]  Uri Zwick,et al.  Improved Rounding Techniques for the MAX 2-SAT and MAX DI-CUT Problems , 2002, IPCO.

[26]  Roded Sharan,et al.  Discovering statistically significant biclusters in gene expression data , 2002, ISMB.

[27]  Roded Sharan,et al.  Revealing modularity and organization in the yeast molecular network by integrated analysis of highly heterogeneous genomewide data. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[28]  R. Steele,et al.  Optimization , 2005, Encyclopedia of Biometrics.