  This paper studies the capacity region of a $K$-user cyclic Gaussian
interference channel, where the $k$th user interferes with only the $(k-1)$th
user (mod $K$) in the network. Inspired by the work of Etkin, Tse and Wang,
which derived a capacity region outer bound for the two-user Gaussian
interference channel and proved that a simple Han-Kobayashi power splitting
scheme can achieve to within one bit of the capacity region for all values of
channel parameters, this paper shows that a similar strategy also achieves the
capacity region for the $K$-user cyclic interference channel to within a
constant gap in the weak interference regime. Specifically, a compact
representation of the Han-Kobayashi achievable rate region using
Fourier-Motzkin elimination is first derived, a capacity region outer bound is
then established. It is shown that the Etkin-Tse-Wang power splitting strategy
gives a constant gap of at most two bits (or one bit per dimension) in the weak
interference regime. Finally, the capacity result of the $K$-user cyclic
Gaussian interference channel in the strong interference regime is also given.
