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 $2/\pi + \Theta(1/{\sqrt n})$, whereas the previous best is $2/\pi + \Theta(1/n)$. We further show a corresponding integrality gap of $2/\pi+\tilde{O}(1/n^{1/3})$.