The positive semidefinite grothendieck problem with rank constraint

Abstract

Given a positive integer n and a positive semidefinite matrix A = (A ij)∈ℝm×m the positive semidefinite Grothendieck problem with rank-n-constraint (SDP n ) is maximize ∑ ∑ Aijxi·xj, where x 1,...,xm∈Sn-1 In this paper we design a randomized polynomial-time approximation algorithm for SDP n achieving an approximation ratio y(n)=2/n (((n+1)/2)/(n/2))2 = 1-O(1/n). of 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 SDP n with a ratio greater than γ(n). We improve the approximation ratio of the best known polynomial-time algorithm for SDP 1 from 2/π to 2/(πγ(m))=2/ π+Θ(1/m), and we show a tighter approximation ratio for SDP n when A is the Laplacian matrix of a weighted graph with nonnegative edge weights. © 2010 Springer-Verlag Berlin Heidelberg.