Abstract : Given a directed weighted graph G, a root r and k terminals, the k-Directed Steiner Tree problem is to find a minimum cost tree rooted at r and spanning all terminals. If this problem has several applications in multicast routing in packet switching networks, the modeling is not adapted anymore in networks based upon the circuit switching principle in which some nodes, called non diffusing nodes, are not able to duplicate packets. We define a more general problem, named Directed Steiner Tree with Limited number of Diffusing nodes (DSTLD), able to model the multicast in a network containing at most d diffusing nodes. We show that DSTLD is XP with respect to d, and use this result to build a ⌈k−1d⌉ -approximation XP in d for DST. Finally, we prove that, under the assumption that NP ⊈ DTIME[n O(loglogn)], there is no polynomial approximation algorithm for DSTLD with ratio 1+(1e−ε)⋅kd−1 for every constant ε > 0.
