  A widely studied problem in communication networks is that of finding the
maximum number of communication requests that can be scheduled concurrently,
subject to node and/or link capacity constraints. In this paper, we consider
the problem of finding the largest number of multicast communication requests
that can be serviced simultaneously by a network of tree topology, subject to
heterogeneous capacity constraints. This problem generalizes the following two
problems studied in the literature: a) the problem of finding a largest induced
$k$-colorable subgraph of a chordal graph, b) the maximum multi-commodity flow
problem in tree networks.
  The problem is already known to be NP-hard and to admit a $c$-approximation
($c \approx 1.58$) in the case of homogeneous capacity constraints. We first
show that the problem is much harder to approximate in the heterogeneous case.
We then use a generalization of a classical algorithm to obtain an
$M$-approximation where $M$ is the maximum number of leaves of the subtrees
representing the multicast communications. Surprisingly, the same algorithm,
though in various disguises, is used in the literature at least four times to
solve related problems (though the analysis is different).
  The special case of the problem where instances are restricted to unicast
communications in a star topology network is known to be polynomial-time
solvable. We extend this result and show that the problem can be solved in
polynomial time for a set of paths in a tree that share a common vertex.
