  In this work, we investigate a multi-source multi-cast network with the aid
of an arbitrary number of relays, where it is assumed that no direct link is
available at each S-D pair. The aim is to find the fundamental limit on the
maximal common multicast throughput of all source nodes if resource allocations
are available. A transmission protocol employing the relaying strategy, namely,
compute-and-forward (CPF), is proposed. {We also adjust the methods in the
literature to obtain the integer network-constructed coefficient matrix (a
naive method, a local optimal method as well as a global optimal method) to fit
for the general topology with an arbitrary number of relays. Two transmission
scenarios are addressed. The first scenario is delay-stringent transmission
where each message must be delivered within one slot. The second scenario is
delay-tolerant transmission where no delay constraint is imposed. The
associated optimization problems to maximize the short-term and long-term
common multicast throughput are formulated and solved, and the optimal
allocation of power and time slots are presented. Performance comparisons show
that the CPF strategy outperforms conventional decode-and-forward (DF)
strategy. It is also shown that with more relays, the CPF strategy performs
even better due to the increased diversity. Finally, by simulation, it is
observed that for a large network in relatively high SNR regime, CPF with the
local optimal method for the network-constructed matrix can perform close to
that with the global optimal method.
