Determining the cells of an arrangement of hyperplanes is a classical problem in combinatorial geometry. In this paper we present an efficient recursive procedure to solve it. In fact, by considering a connection between the latter problem and optimality conditions of unconstrained quadratic optimization problems in the following form: (QP) min{xQx | x in {-1,1}n}, with Q in Rn×n, we can show the proposed algorithm solves problems of the form (QP) in polynomial time when the rank of the matrix Q is fixed and the number of its positive diagonal entries is O(log(n)).