2 independent message, one for each other user. Thus, we modify the “functional decode-and-forward" scheme accordingly to obtain a lower bound for the sum-capacity using lattice alignment. Other lower bounds are obtained by using complete decode and forward, or by operating the Y-channel as a bi-directional relay channel where only two users are active at the same time.

Comparing the upper bounds and lower bounds, we bound the gap between them for the case of equal power at all nodes. This gap is shown to be less than 5/2 bits for all values of channel coefficients. Moreover, this gap is shown to approach 3/2 bits as power increases. We also bound the multiplicative gap between the bounds by 3. For the symmetric Y-channel where all channel gains are equal, we show that the gap between these bounds is less than one bit.

For the restricted Y-channel, the bounds are further tightened, and we characterize the sum-capacity within 2 bits for all values of channel coefficients when all nodes have equal power. This gap is shown to approach 1 bit as power increases.

The rest on the paper is organized as follows. The system model is described in section \[Model\]. The 