  Intelligent reflecting surface (IRS) is envisioned to play a significant role
in future wireless communication systems as an effective means of reconfiguring
the radio signal propagation environment. In this paper, we study a new
multi-IRS aided massive multiple-input multiple-output (MIMO) system, where a
multi-antenna BS transmits independent messages to a set of remote
single-antenna users using orthogonal beams that are subsequently reflected by
different groups of IRSs via their respective multi-hop passive beamforming
over pairwise line-of-sight (LoS) links. We aim to select optimal IRSs and
their beam routing path for each of the users, along with the active/passive
beamforming at the BS/IRSs, such that the minimum received signal power among
all users is maximized. This problem is particularly difficult to solve due to
a new type of path separation constraints for avoiding the IRS-reflected signal
induced interference among different users. To tackle this difficulty, we first
derive the optimal BS/IRS active/passive beamforming solutions based on their
practical codebooks given the reflection paths. Then we show that the resultant
multi-beam multi-hop routing problem can be recast as an equivalent
graph-optimization problem, which is however NP-complete. To solve this
challenging problem, we propose an efficient recursive algorithm to partially
enumerate the feasible routing solutions, which is able to effectively balance
the performance-complexity trade-off. Numerical results demonstrate that the
proposed algorithm achieves near-optimal performance with low complexity and
outperforms other benchmark schemes. Useful insights into the optimal
multi-beam multi-hop routing design are also drawn under different setups of
the multi-IRS aided massive MIMO network.
