Efficient computation of the binary vector that maximizes a rank-deficient quadratic form

The maximization of a full-rank quadratic form over a finite alphabet is NP-hard in both a worst-case sense and an average sense. Interestingly, if the rank of the form is not a function of the problem size, then it can be maximized in polynomial time. An algorithm for the efficient computation of the binary vector that maximizes a rank-deficient quadratic form is developed based on an analytic procedure. Auxiliary spherical coordinates are introduced and the multi-dimensional space is partitioned into a polynomial-size set of regions; each region corresponds to a distinct binary vector. The binary vector that maximizes the rank-deficient quadratic form is shown to belong to the polynomial-size set of candidate vectors. Thus, the size of the feasible set is efficiently reduced from exponential to polynomial.