  The two-user Gaussian interference channel with a shared out-of-band relay is
considered. The relay observes a linear combination of the source signals and
broadcasts a common message to the two destinations, through a perfect link of
fixed limited rate $R_0$ bits per channel use. The out-of-band nature of the
relay is reflected by the fact that the common relay message does not interfere
with the received signal at the two destinations. A general achievable rate is
established, along with upper bounds on the capacity region for the Gaussian
case. For $R_0$ values below a certain threshold, which depends on channel
parameters, the capacity region of this channel is determined in this paper to
within a constant gap of $\Delta=1.95$ bits. We identify interference regimes
where a two-for-one gain in achievable rates is possible for every bit relayed,
up to a constant approximation error. Instrumental to these results is a
carefully-designed quantize-and-forward type of relay strategy along with a
joint decoding scheme employed at destination ends. Further, we also study
successive decoding strategies with optimal decoding order (corresponding to
the order at which common, private, and relay messages are decoded), and show
that successive decoding also achieves two-for-one gains asymptotically in
regimes where a two-for-one gain is achievable by joint decoding; yet,
successive decoding produces unbounded loss asymptotically when compared to
joint decoding, in general.
