  This paper considers a fully-connected interference network with a relay in
which multiple users equipped with a single antenna want to exchange multiple
unicast messages with other users in the network by sharing the relay equipped
with multiple antennas. For such a network, the degrees of freedom (DoF) are
derived by considering various message exchange scenarios: a multi-user
fully-connected Y channel, a two-pair two-way interference channel with the
relay, and a two-pair two-way X channel with the relay. Further, considering
distributed relays employing a single antenna in the two-way interference
channel and the three-user fully-connected Y channel, achievable sum-DoF are
also derived in the two-way interference channel and the three-user
fully-connected Y channel. A major implication of the derived DoF results is
that a relay with multiple antennas or multiple relays employing a single
antenna increases the capacity scaling law of the multi-user interference
network when multiple directional information flows are considered, even if the
networks are fully-connected and all nodes operate in half-duplex. These
results reveal that the relay is useful in the multi-way interference network
with practical considerations.
