A sub-constant improvement in approximating the positive semidefinite Grothendieck problem

Semidefinite relaxations are a powerful tool for approximately solving combinatorial optimization problems such as MAX-CUT and the Grothendieck problem. By exploiting a bounded rank property of extreme points in the semidefinite cone, we make a sub-constant improvement in the approximation ratio of one such problem. Precisely, we describe a polynomial-time algorithm for the positive semidefinite Grothendieck problem – based on rounding from the standard relaxation – that achieves a ratio of 2/π + Θ(1/ √ n), whereas the previous best is 2/π + Θ(1/n). We further show a corresponding integrality gap of 2/π + Õ(1/n1/3).