  In this paper we study the multisource multicast problem where every sink in
a given directed acyclic graph is a client and is interested in a common file.
We consider the case where each node can have partial knowledge about the file
as a side information. Assuming that nodes can communicate over the capacity
constrained links of the graph, the goal is for each client to gain access to
the file, while minimizing some linear cost function of number of bits
transmitted in the network. We consider three types of side-information
settings:(ii) side information in the form of linearly correlated packets; and
(iii) the general setting where the side information at the nodes have an
arbitrary (i.i.d.) correlation structure. In this work we 1) provide a
polynomial time feasibility test, i.e., whether or not all the clients can
recover the file, and 2) we provide a polynomial-time algorithm that finds the
optimal rate allocation among the links of the graph, and then determines an
explicit transmission scheme for cases (i) and (ii).
