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

Roy Frostig; Sida I. Wang

Paper

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

Abstract

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

, whereas the previous best is

. We further show a corresponding integrality gap of