This paper studies a Gaussian relay network, where the relays can either transmit or receive at any given time, but not both. Known upper (cut-set) and lower (noisy network coding) bounds on the capacity of a memoryless full-duplex relay network are specialized to the half-duplex case and shown to be to within a constant gap of one another. For fairly broad range of relay network sizes, the derived gap is smaller than what is known in the literature, and it can be further reduced for more structured networks such as diamond networks. It is shown that the asymptotically optimal duration of the listen and transmit phases for the relays can be obtained by solving a linear program; the coefficients of the linear constraints of this linear program are the solution of certain 'assignment problems' for which efficient numerical routines are available; this gives a general interesting connection between the high SNR approximation of the capacity of a MIMO channel and the 'assignment problem' in graph theory. Finally, some results available for diamond networks are extended to general networks. For a general relay network with 2 relays, it is proved that, out of the 4 possible listen/transmit states, at most 3 have a strictly positive probability. Numerical results for a network with K - 2 < 9 relays show that at most K-1 states have a strictly positive probability, which is conjectured to be true for any number of relays.