  In ISIT'12 Brahma, \"{O}zg\"{u}r and Fragouli conjectured that in a
half-duplex diamond relay network (a Gaussian noise network without a direct
source-destination link and with $N$ non-interfering relays) an approximately
optimal relay scheduling (achieving the cut-set upper bound to within a
constant gap uniformly over all channel gains) exists with at most $N+1$ active
states (only $N+1$ out of the $2^N$ possible relay listen-transmit
configurations have a strictly positive probability). Such relay scheduling
policies are said to be simple. In ITW'13 we conjectured that simple relay
policies are optimal for any half-duplex Gaussian multi-relay network, that is,
simple schedules are not a consequence of the diamond network's sparse
topology. In this paper we formally prove the conjecture beyond Gaussian
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 optimal schedule exists with at most $N+1$
active states. The key step of our proof is to write the minimum of a
submodular function by means of its Lov\'{a}sz extension and use the greedy
algorithm for submodular polyhedra to highlight structural properties of the
optimal solution. This, together with the saddle-point property of min-max
problems and the existence of optimal basic feasible solutions in linear
programs, proves the claim.
