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 on average 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 only a linear number of circular arcs that possibly contain the optimal point. Furthermore, the proposed algorithm can be viewed as a novel approach to a special case of the discrete quadratic program (QP).

翻译：在多项式时间内找到智能反射面（IRS）离散约束下相移的最优配置仍然是一个悬而未决的问题。上述问题被广泛认为十分困难，因为它与任何已知可高效求解的组合问题均无关联。分支定界算法和近似算法构成了该领域的最佳成果。然而，本研究表明，就IRS反射单元（RE）数量而言，全局最优解实际上可在平均线性时间内达到。其主要思想是将离散波束成形问题几何解释为选择单位圆上的最优点。尽管相移的可能组合数随RE数量呈指数增长，但结果证明，可能包含最优点的圆弧数量仅为线性。此外，所提算法可视为离散二次规划（QP）特例的一种新颖求解方法。