Approximating the Little Grothendieck Problem over the Orthogonal Group
Afonso S. Bandeira, Christopher Kennedy, Amit Singer
The little Grothendieck problem (a special case of Boolean quadratic optimization) consists of maximizing P ij Cijxixj over binary variables xi 2 f 1g, where C is a positive semidenite matrix. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C 2 R dn dn a positive semidenite matrix, the objective is to maximize P ij tr C T ijOiO T restricting Oi to take values in the group of orthogonal matrices O(d), where Cij denotes the (ij)-th d d block of C. We propose an approximation algorithm, to which we refer as Orthogonal-Cut, to solve the little Grothendieck problem over the group of orthogonal matrices O(d) and show a constant approximation ratio. Our method is based on semidenite programming where the relaxation is inspired by the work of Goemans and Williamson in the context of the MaxCut problem. For a given d 1, we show a constant approximation ratio of 2, where d is the expected average singular value of a d d matrix with random GaussianN 0; 1 i.i.d. entries. For d = 1 we recover the known 2 = 2= approximation guarantee for the classical little Grothendieck problem. Orthogonal-Cut also serves as an approximation algorithm for several applications including the Procrustes problem where it improves over the best previously known approximation ratio of 1 2 p 2 . The little Grothendieck problem falls under the larger class of problems approximated by an algorithm recently proposed in the context of the non-commutative Grothendieck inequality. Nonetheless, our approach is simpler and gives a better approximation ratio.
•Aug 23, 2013•7 citations