  In this paper, we study an intelligent reflecting surface (IRS) assisted
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). The IRS
phase shifts are adjusted so as to maximize the ideal SNR (i.e., without CSI
error), based only on the estimated channels. First, we derive a closed-form
expression of the worst-case SNR, and then formulate the robust (discrete)
optimization problem. Moreover, we design and analyze 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. Finally, numerical simulations confirm the theoretical results. In
particular, the proposed algorithm shows identical performance with the
exhaustive search, and significantly outperforms a baseline scheme, namely, the
activation of all IRS elements.
