The Positive Semidefinite Grothendieck Problem with Rank Constraint

Given a positive integer n and a positive semidefinite matrix A = (Aij) ∈ Rm × m the positive semidefinite Grothendieck problem with rank-n-constraint (SDPn) is maximize Σi=1m Σj=1m Aij xi ċ xj, where x1, ..., xm ∈ Sn-1. In this paper we design a randomized polynomial-time approximation algorithm for SDPn achieving an approximation ratio of γ(n) = 2/n(Γ((n + 1)/2)/Γ(n/2))2 = 1 - Θ(1/n). We show that under the assumption of the unique games conjecture the achieved approximation ratio is optimal: There is no polynomial-time algorithm which approximates SDPn with a ratio greater than γ(n). We improve the approximation ratio of the best known polynomial-time algorithm for SDP1 from 2/π to 2/(πγ(m)) = 2/π + Θ(1/m), and we show a tighter approximation ratio for SDPn when A is the Laplacian matrix of a weighted graph with nonnegative edge weights.

[1]  Arkadi Nemirovski,et al.  On Tractable Approximations of Uncertain Linear Matrix Inequalities Affected by Interval Uncertainty , 2002, SIAM J. Optim..

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

[3]  Leon M. Hall,et al.  Special Functions , 1998 .

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

[5]  H. Buhrman,et al.  A generalized Grothendieck inequality and entanglement in XOR games , 2009, 0901.2009.

[6]  Subhash Khot,et al.  Approximate Kernel Clustering , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[7]  Guy Kindler,et al.  Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

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

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

[10]  I. J. Schoenberg.  Positive definite functions on spheres , 1942 .

[11]  H. E. Stanley,et al.  Spherical Model as the Limit of Infinite Spin Dimensionality , 1968 .

[12]  Subhash Khot,et al.  Sharp kernel clustering algorithms and their associated Grothendieck inequalities , 2009, SODA '10.

[13]  Noga Alon,et al.  Quadratic forms on graphs , 2005, STOC '05.

[14]  Subhash Khot,et al.  Sharp kernel clustering algorithms and their associated Grothendieck inequalities , 2009, ACM-SIAM Symposium on Discrete Algorithms.

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

[16]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[17]  Noga Alon,et al.  Approximating the cut-norm via Grothendieck's inequality , 2004, STOC '04.

[18]  Subhash Khot,et al.  Vertex cover might be hard to approximate to within 2-/spl epsiv/ , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[19]  George Karakostas,et al.  A better approximation ratio for the vertex cover problem , 2005, TALG.

[20]  Nathan Linial,et al.  Lower bounds in communication complexity based on factorization norms , 2007, STOC '07.

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

[22]  KarakostasGeorge.  A better approximation ratio for the vertex cover problem , 2009 .

[23]  Prasad Raghavendra,et al.  Towards computing the Grothendieck constant , 2009, SODA.

[24]  Subhash Khot,et al.  On the power of unique 2-prover 1-round games , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[25]  Prasad Raghavendra,et al.  Optimal algorithms and inapproximability results for every CSP? , 2008, STOC.

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