  Intelligent reflecting surface (IRS) is envisioned to play a significant role
in future wireless communication systems thanks to its powerful capability of
enabling smart and reconfigurable radio environment. In this paper, we study
the multi-IRS aided downlink communication in a massive multiple-input
multiple-output (MIMO) system, where a multi-antenna BS simultaneously serves
multiple remote single-antenna users with orthogonal beams reflected by
multiple IRSs. By exploiting the line-of-sight (LoS) link between each pair of
selected IRSs, a multi-hop cascaded LoS link can be established between the BS
and each user via their cooperative beam routing. Under this setup, we optimize
the selected IRSs and their beam routing path for each user, along with the
BS/IRS active/passive beamforming such that the minimum received signal power
among all users is maximized, subject to a new multi-beam routing path
separation constraint for avoiding the inter-user/route interference. To tackle
this problem, we first derive the optimal BS/IRS active/passive beamforming in
closed-form for any given beam routes and show the beam routing optimization is
NP-complete by recasting it as an equivalent graph-optimization problem. To
solve this challenging problem, we then propose an efficient recursive
algorithm to partially enumerate the feasible solutions, which effectively
balances the performance-complexity trade-off by tuning its design parameter.
Numerical results demonstrate that the proposed algorithm can achieve
near-optimal performance with low enumeration complexity and also outperform
other benchmark schemes.
