  Intelligent reflecting surfaces (IRSs) have emerged as a promising wireless
technology for the dynamic configuration and control of electromagnetic waves,
thus creating a smart (programmable) radio environment. In this context, we
study a multi-IRS assisted two-way communication system consisting of two users
that employ full-duplex (FD) technology. More specifically, we deal with the
joint IRS location and size (i.e., the number of reflecting elements)
optimization in order to minimize an upper bound of system outage probability
under various constraints: minimum and maximum number of reflecting elements
per IRS, maximum number of installed IRSs, maximum total number of reflecting
elements (implicit bound on the signaling overhead) as well as maximum total
IRS installation cost. First, the problem is formulated as a discrete
optimization problem and, then, a theoretical proof of its NP-hardness is
given. Moreover, we provide a lower bound on the optimum value by solving a
linear-programming relaxation (LPR) problem. Subsequently, we design two
polynomial-time algorithms, a deterministic greedy algorithm and a randomized
approximation algorithm, based on the LPR solution. The former is a heuristic
method that always computes a feasible solution for which (a posteriori)
performance guarantee can be provided. The latter achieves an approximate
solution, using randomized rounding, with provable (a priori) probabilistic
guarantees on the performance. Furthermore, extensive numerical simulations
demonstrate the superiority of the proposed algorithms compared to the baseline
schemes. Finally, useful conclusions regarding the comparison between FD and
conventional half-duplex (HD) systems are also drawn.
