  In this paper, we study the amplify-and-forward (AF) schemes in two-hop
one-way relay networks. In particular, we consider the multigroup multicast
transmission between long-distance users. Given that perfect channel state
information is perceived, our goal is to design the AF process so that the
max-min-fair (MMF) signal-to-interference-plus-noise ratio (SINR) is optimized
subject to generalized power constraints. We propose a rank-two beamformed
Alamouti (BFA) AF scheme and formulate the corresponding AF design problem as a
\emph{two-variable} fractional quadratically-constrained quadratic program
(QCQP), which is further tackled by the semidefinite relaxation (SDR)
technique. We analyze the approximation quality of two-variable fractional SDRs
under the Gaussian randomization algorithm. These results are fundamentally new
and reveal that the proposed BFA AF scheme can outperform the traditional BF AF
scheme, especially when there are many users in the system or many generalized
power constraints in the problem formulation. From a practical perspective, the
BFA AF scheme offers two degrees of freedom (DoFs) in beamformer design, as
opposed to the one DoF offered by the BF AF scheme, to improve the receivers'
SINR. In the latter part of this paper, we demonstrate how this extra DoF leads
to provable performance gains by considering two special cases of multicasting,
where the AF process is shown to employ a special structure. The numerical
simulations further validate that the proposed BFA AF scheme outperforms the BF
AF scheme and works well for large-scale relay systems.
