Solving the fixed rank convex quadratic maximization in binary variables by a parallel zonotope construction algorithm

Abstract We address the weighted max-cut problem, or equivalently the problem of maximizing a quadratic form in n binary variables. If the underlying (symmetric) matrix is positive semidefinite of fixed rank d , then the problem can be reduced to searching the extreme points of a zonotope, thus becoming of polynomial complexity in O( n d  − 1 ). Reverse search is an efficient and practical means for enumerating the cells of a regular hyperplane arrangement, or equivalently, the extreme points of a zonotope. We present an enhanced version of reverse search of significantly reduced computational complexity that uses ray shooting and is well suited for parallel computation. Furthermore, a neighborhood zonotope edge following descent heuristic can be devised. We report preliminary computational experiments of a parallel implementation of our algorithms.