  This paper considers a general Gaussian relay network where a source
transmits a message to a destination with the help of N half-duplex relays. It
proves that the information theoretic cut-set upper bound to the capacity can
be achieved to within 2:021(N +2) bits with noisy network coding, thereby
reducing the previously known gap. Further improved gap results are presented
for more structured networks like diamond networks. It is then shown that the
generalized Degrees-of-Freedom of a general Gaussian half-duplex relay network
is the solution of a linear program, where the coefficients of the linear
inequality constraints are proved to be the solution of several linear
programs, known in graph theory as the assignment problem, for which efficient
numerical algorithms exist. The optimal schedule, that is, the optimal value of
the 2^N possible transmit-receive configurations/states for the relays, is
investigated and known results for diamond networks are extended to general
relay networks. It is shown, for the case of 2 relays, that only 3 out of the 4
possible states have strictly positive probability. Extensive experimental
results show that, for a general N-relay network with N<9, the optimal schedule
has at most N +1 states with strictly positive probability. As an extension of
a conjecture presented for diamond networks, it is conjectured that this result
holds for any HD relay network and any number of relays. Finally, a 2-relay
network is studied to determine the channel conditions under which selecting
the best relay is not optimal, and to highlight the nature of the rate gain due
to multiple relays.
