  In this paper, a dual-hop communication system composed of a source S and a
destination D connected through two non-interfering half-duplex relays, R1 and
R2, is considered. In the literature of Information Theory, this configuration
is known as the diamond channel. In this setup, four transmission modes are
present, namely: 1) S transmits, and R1 and R2 listen (broadcast mode), 2) S
transmits, R1 listens, and simultaneously, R2 transmits and D listens. 3) S
transmits, R2 listens, and simultaneously, R1 transmits and D listens. 4) R1,
R2 transmit, and D listens (multiple-access mode). Assuming a constant power
constraint for all transmitters, a parameter $\Delta$ is defined, which
captures some important features of the channel. It is proven that for
$\Delta$=0 the capacity of the channel can be attained by successive relaying,
i.e, using modes 2 and 3 defined above in a successive manner. This strategy
may have an infinite gap from the capacity of the channel when $\Delta\neq$0.
To achieve rates as close as 0.71 bits to the capacity, it is shown that the
cases of $\Delta$>0 and $\Delta$<0 should be treated differently. Using new
upper bounds based on the dual problem of the linear program associated with
the cut-set bounds, it is proven that the successive relaying strategy needs to
be enhanced by an additional broadcast mode (mode 1), or multiple access mode
(mode 4), for the cases of $\Delta$<0 and $\Delta$>0, respectively.
Furthermore, it is established that under average power constraints the
aforementioned strategies achieve rates as close as 3.6 bits to the capacity of
the channel.
