  In this paper, we study an intelligent reflecting surface (IRS)-aided
communication system with single-antenna transmitter and receiver, under
imperfect channel state information (CSI). More specifically, we deal with the
robust selection of binary (on/off) states of the IRS elements in order to
maximize the worst-case energy efficiency (EE), given a bounded CSI
uncertainty, while satisfying a minimum signal-to-noise ratio (SNR). In
addition, we consider not only continuous but also discrete IRS phase shifts.
First, we derive closed-form expressions of the worst-case SNRs, and then
formulate the robust (discrete) optimization problems for each case. In the
case of continuous phase shifts, we design a dynamic programming (DP) algorithm
that is theoretically guaranteed to achieve the global maximum with polynomial
complexity $O(L\,{\log L})$, where $L$ is the number of IRS elements. In the
case of discrete phase shifts, we develop a convex-relaxation-based method
(CRBM) to obtain a feasible (sub-optimal) solution in polynomial time
$O(L^{3.5})$, with a posteriori performance guarantee. Furthermore, numerical
simulations provide useful insights and confirm the theoretical results. In
particular, the proposed algorithms are several orders of magnitude faster than
the exhaustive search when $L$ is large, thus being highly scalable and
suitable for practical applications. Moreover, both algorithms outperform a
baseline scheme, namely, the activation of all IRS elements.
