It remains an open problem to find the optimal configuration of phase shifts
under the discrete constraint for intelligent reflecting surface (IRS) in
polynomial time. The above problem is widely believed to be difficult because
it is not linked to any known combinatorial problems that can be solved
efficiently. The branch-and-bound algorithms and the approximation algorithms
constitute the best results in this area. Nevertheless, this work shows that
the global optimum can actually be reached in linear time in terms of the
number of reflective elements (REs) of IRS. The main idea is to geometrically
interpret the discrete beamforming problem as choosing the optimal point on the
unit circle. Although the number of possible combinations of phase shifts grows
exponentially with the number of REs, it turns out that there are merely a
linear number of points on the unit circle to consider. Furthermore, the
proposed algorithm can be viewed as a novel approach to a special case of the
discrete quadratic program (QP).Comment: 5 page