A Linear Time Algorithm for the Optimal Discrete IRS Beamforming

TL;DR

This paper introduces a linear time algorithm to find the globally optimal discrete IRS beamforming configuration by geometrically interpreting the problem and reducing the search space to a linear number of candidate arcs.

Contribution

It presents the first linear time algorithm for optimal discrete IRS beamforming, significantly improving over existing exponential or polynomial methods.

Findings

Global optimum achievable in linear time on average

Reduces search to a linear number of circular arcs

Applicable as a novel approach to discrete quadratic programming

Abstract

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).