This paper studies networks with N half-duplex relays assisting the communication between a source and a destination. In ISIT'12 Brahma, Özgür and Fragouli conjectured that in Gaussian half-duplex diamond networks (i.e., without a direct link between the source and the destination, and with N non-interfering relays) an approximately optimal relay scheduling policy (i.e., achieving the cut-set upper bound to within a constant gap) has at most N+1 active states (i.e., at most N+1 out of the $2^N$ possible relay listen-transmit states have a strictly positive probability). Such relay scheduling policies were referred to as simple. In ITW'13 we conjectured that simple approximately optimal relay scheduling policies exist for any Gaussian half-duplex multi-relay network irrespectively of the topology. This paper formally proves this more general version of the conjecture and shows it holds beyond Gaussian noise networks. In particular, for any memoryless half-duplex N-relay network with independent noises and for which independent inputs are approximately optimal in the cut-set upper bound, an approximately optimal simple relay scheduling policy exists. A convergent iterative polynomial-time algorithm, which alternates between minimizing a submodular function and maximizing a linear program, is proposed to find the approximately optimal simple relay schedule. As an example, for N-relay Gaussian networks with independent noises, where each node in equipped with multiple antennas and where each antenna can be configured to listen or transmit irrespectively of the others, the existence of an approximately optimal simple relay scheduling policy with at most N+1 active states is proved. Through a line-network example it is also shown that independently switching the antennas at each relay can provide a strictly larger multiplexing gain compared to using the antennas for the same purpose.